Model-based method for determining the road handling performance of a tyre of a wheel for a vehicle

ABSTRACT

A method for determining the road handling of a tire, comprising descriptions of the tire by means of a first, concentrated-parameter, physical model and by means of a second, finite-element model, a simulation on the second, finite-element model of a selected series of dynamic tests and an application to the first physical model of equations of motion suitable for representing the dynamic tests in order to obtain first and second frequency responses of selected quantities; a comparison between the first and second frequency responses of the selected quantities for determining the concentrated parameters of the first physical model and physical quantities indicative of the drift behavior of the tire.

[0001] This application is based on European Patent Application No.98830209.7 filed on Apr. 7, 1998 and U.S. Provisional Application No.60/092,594 filed on Jul. 10, 1998, the content of which is incorporatedhereinto by reference.

[0002] This invention relates to a method for determining the roadhandling of a tire of a wheel for a vehicle.

[0003] At the present time, to determine the road handling performanceof a tire, the manufacturers of pneumatic tires are obliged to producenumerous physical prototypes in order to experimentally evaluate theeffects of the various design parameters on the drift behaviour of thetire, under steady state and transient state conditions. Theexperimental tests are conducted according to iterative procedures, thatare largely empirical and based on experience and are also extremelydemanding in terms of time and cost.

[0004] Furthermore, automobile manufacturing companies are insistingmore and more frequently that the makers of pneumatic tires come up withtires with extremely precise technical characteristics as early as theinitial stages of vehicle study and dynamic behaviour forecasting.

[0005] In such a position, the tire manufacturers are finding it verydifficult to respond satisfactorily and with the necessary flexibilityto the various market demands.

[0006] The object of this invention is to provide a scientificmethodology with which to identify the performance characteristics of atire in relation to road handling, on the basis of previously defineddesign specifications.

[0007] The above object is achieved according to this invention by amethod for determining the road handling of a tire of a wheel for avehicle, said tire being made from selected mixes of rubber andreinforcing materials, said method comprising:

[0008] a) a first description of said tire by means of a first,concentrated-parameter, physical model, said first physical modelcomprising a rigid ring which represents the tread band provided withinserts, a belting structure and corresponding carcass portion of saidtire, a disk which represents a hub of said wheel and beading of saidtire, principal springs and dampers connecting said rigid ring to saidhub and representing sidewalls of said tire and air under pressureinside said tire, supplementary springs and dampers representingdeformation phenomena of said belting structure through the effect of aspecified vertical load, and a brush model simulating physical phenomenain an area of contact between said tire and a road, said area of contacthaving a dynamic length 2a,

[0009] b) a definition of selected degrees of freedom of said firstphysical model, and

[0010] c) an identification of equations of motion suitable fordescribing the motion of said first physical model under selecteddynamic conditions, characterized in that it comprises

[0011] d) the definition of said concentrated parameters, saidconcentrated parameters consisting of the mass M_(c) and a diametralmoment of inertia J_(c) of said rigid ring, the mass M_(m) and adiametral moment of inertia J_(m) of said disk, structural stiffnessesK_(c) and structural dampings R_(c) respectively of said principalsprings and dampers, and residual stiffnesses K_(r) and residualdampings R_(r) respectively of said supplementary springs and dampers,wherein

[0012] said structural stiffnesses K_(c) consist of lateral stiffnessK_(cy) between said hub and said belt, camber torsional stiffnessK_(cθx) between said hub and said belt and yawing torsional stiffnessK_(cθz) between said hub and said belt,

[0013] said structural dampings R_(c) consist of lateral damping R_(cy)between said hub and said belt, camber torsional damping R_(cθx) betweensaid hub and said belt and yawing torsional damping R_(cθz) between saidhub and said belt,

[0014] said residual stiffnesses K_(r) consist of residual lateralstiffness K_(ry), residual camber torsional stiffness K_(rθx) andresidual yawing torsional stiffness K_(rθz), and

[0015] said residual dampings R_(r) consist of residual lateral dampingR_(ry), residual camber torsional damping R_(rθx) and residual yawingtorsional damping R_(rθz),

[0016] e) a description of said tire by means of a second,finite-element model comprising first elements with a selected number ofnodes, suitable for describing said mixes, and second elements suitablefor describing said reinforcing materials, each first finite elementbeing associated with a first stiffness matrix which is determined bymeans of a selected characterization of said mixes and each secondelement being associated with a second supplementary stiffness matrixwhich is determined by means of a selected characterization of saidreinforcing materials,

[0017] f) a simulation on said second, finite-element model of aselected series of virtual dynamic tests for exciting said second modelaccording to selected procedures and obtaining transfer functions andfirst frequency responses of selected quantities, measured at selectedpoints of said second model,

[0018] g) a description of the behaviour of said first physical model bymeans of equations of motion suitable for representing the above dynamictests for obtaining second frequency responses of said selectedquantities, measured at selected points of said first physical model,

[0019] h) a comparison between said first and said second frequencyresponses of said selected quantities to determine errors that are afunction of said concentrated parameters of said first physical model,and

[0020] i) the identification of values for said concentrated parametersthat minimize said errors so that said concentrated parameters describethe dynamic behaviour of said tire,

[0021] j) the determination of selected physical quantities suitable forindicating the drift behaviour of said tire, and

[0022] k) the evaluation of the drift behaviour of said tire by means ofsaid physical quantities.

[0023] To advantage, said selected physical quantities are the totaldrift stiffness K_(d) of said tire, in turn comprising the structuralstiffness K_(c) and the tread stiffness K_(b), and the total camberstiffness K_(γ) of said tire.

[0024] According to a preferred embodiment, said method also comprises

[0025] l) a definition of said brush model, said brush model having astiffness per unit of length c_(py) and comprising at least one rigidplate, at least one deformable beam having a length equal to the length2a of said area of contact and at least one microinsert associated withsaid beam, said microinsert consisting of at least one set of springsdistributed over the entire length of said beam, said springsreproducing the uniformly distributed, lateral and torsional stiffnessof said area of contact.

[0026] Preferably, said degrees of freedom referred to at previous pointb) are composed of:

[0027] absolute lateral displacement y_(m) of said hub, absolute yawrotation σ_(m) of said hub and absolute rolling rotation ρ_(m) of saidhub,

[0028] relative lateral displacement y_(c) of said belt with respect tosaid hub, relative yaw rotation σ_(c) of said belt with respect to saidhub and relative rolling rotation ρ_(c) of said belt with respect tosaid hub,

[0029] absolute lateral displacement y_(b) of said plate, absolute yawrotation σ_(b) of said plate and absolute rolling rotation ρ_(b) of saidplate, and

[0030] absolute lateral displacement y_(s) of the bottom ends of said atleast one microinsert.

[0031] According to another embodiment, said selected series of virtualdynamic tests referred to at previous point f) comprises a first and asecond test with said tire blown up and not pressed to the ground, saidfirst test consisting in imposing a translation in the transversedirection y on the hub and in measuring the lateral displacement y_(c)of at least one selected cardinal point of said belt and the forcecreated between said hub and said belt in order to identify said massM_(c), said lateral stiffness K_(cy), and said lateral damping R_(cy),said second test consisting in imposing a camber rotation θ_(x) on saidhub and in measuring the lateral displacement of at least one selectedcardinal point of said belt y_(c) and the torque transmitted betweensaid hub and said belt in order to identify said diametral moment ofinertia J_(c), said camber torsional stiffness K_(cθx), said cambertorsional damping R_(cθx), said yawing torsional stiffness K_(cθz) andsaid yawing torsional damping R_(cθz).

[0032] Preferably said selected series of virtual dynamic tests referredto at previous point f) also comprises a third and a fourth test withsaid tire blown up, pressed to the ground and bereft of said tread atleast in said area of contact, said third test consisting in applying tosaid hub a sideward force in the transverse direction F_(y) and inmeasuring the lateral displacement y_(c) of said hub and of at least twoselected cardinal points of said belt in order to identify said residuallateral stiffness K_(ry), said residual lateral damping R_(ry), saidcamber residual stiffness K_(rθx), and said camber residual dampingR_(rθx), said fourth test consisting in applying to said hub a yawingtorque C_(θz) and in measuring the yaw rotation of said hub σ_(m) andthe lateral displacement y_(c) of at least one selected cardinal pointof said belt in order to identify said residual yawing stiffness K_(rθz)and said residual yawing damping R_(rθz).

[0033] According to another embodiment, said method also comprises

[0034] m) an application to said first physical model of a drift angleα, starting from a condition in which said at least one beam is in anon-deformed configuration and said brush model has a null snakingσ_(b),

[0035] n) the determination of the sideward force and the self-aligningtorque that act on said hub through the effect of said drift α and whichdepend on the difference α−σ_(b) and on the deformation of said at leastone beam,

[0036] o) the determination of the deformation curve of said at leastone beam,

[0037] p) an application of said sideward force and said self-aligningtorque to said second, finite-element model in order to obtain apressure distribution on said area of contact and

[0038] q) the determination of the sideward force and the self-aligningtorque that act on said hub through the effect of said drift on saidfirst physical model, that depend on the pressure distributioncalculated in the previous step p),

[0039] r) a check, by means of said pressure distribution obtained inthe previous step p), that said sideward force and said self-aligningtorque are substantially similar to those calculated in previous stepq),

[0040] s) a determination of the sideward force and of the self-aligningtorque for said angle of drift, and

[0041] t) repetition of the procedure from step m) to step s) fordifferent values of the drift angle α to obtain drift, force andself-alignment torque curves, suitable for indicating the driftbehaviour under steady state conditions of said tire, and

[0042] u) the evaluation of the steady state drift behaviour of saidtire.

[0043] According to another preferred embodiment, said method alsocomprises:

[0044] i) a simulation of the behaviour of said first physical model inthe drift transient state by means of equations of motion reproducingselected experimental drift tests, and

[0045] ii) the determination, with a selected input of a steering angleimposed on said hub, of the pattern with time of the selected freedegrees of freedom of said first physical model, of the sideward forceand of the self-aligning torque in said area of contact in order todetermine the length of relaxation of said tire.

[0046] To advantage, said first elements of said second, finite-elementmodel have linear form functions and their stiffness matrix isdetermined by means of selected static and dynamic tests conducted onspecimens of said mixes, whereas the stiffness matrix of said secondelements is determined by means of selected static tests on specimens ofsaid reinforcing materials.

[0047] With the method according to this invention, three main resultsare obtained:

[0048] 1. determination of the links between physical parameters of thetire and its structural properties;

[0049] 2. determination of the steady state drift behaviour of the tire,without the need to build prototypes at this stage;

[0050] 3. determination of the transient state behaviour of the tire,when a generic law of motion is imposed on the hub, without the need tobuild prototypes at this stage.

[0051] These results have been achieved by the production of a verysimple, first physical model, with only nine degrees of freedom, thatmanages to make allowance for the majority of the structuralcharacteristics of the actual tire.

[0052] The structural characteristics of the tire are reproduced in thefirst physical model by means of an appropriate condensation ofconcentrated equivalent masses, stiffnesses and dampings.

[0053] In practice, the concentrated-parameter model is equivalent to akind of dynamic concentration of the complex finite-element model,summarizing all its dynamic characteristics in a low number of mass,damping and stiffness parameters.

[0054] More specifically, it has been proven that this correspondencemay be held valid in the range of frequencies between 0 and 80 Hz.

[0055] The method enables identification of the structural parametersneeded for the complete description of the first physical model usingsimulated virtual numerical tests with a second, extremely detailedmodel built from finite element models (F.E.M.) reproducing thebehaviour of the non-rolling tire (not drifting).

[0056] One of the major advantages of the method according to theinvention is that it partly dispenses with the need to constructphysical prototypes and the resultant experimental tests in theiterative process of tire determination, replacing this approach withvirtual prototyping.

[0057] The tire's design parameters (characteristics of the mixes,inclination of the threads, shape of the sidewalls, width of the belt,etc.) are directly fed into the second, finite-element model which isextremely detailed.

[0058] The concentrated parameters of the first physical model areidentified by minimizing the difference between the vibrational dynamicbehaviour of the second, finite-element model of the non-rolling tireand the corresponding response given by the first physical model.

[0059] The identification procedure defined comprises various operationsthat are executed in a precise, pre-established order. Starting from thetransfer functions obtained by means of a series of virtual dynamictests conducted on the non-rolling finite-element model, the masses,stiffnesses and dampings of the concentrated-parameter model aredetermined, providing a better description of the dynamic behaviour ofthe tire.

[0060] Thus the identification procedure enables a link to beestablished between the design parameters (fed into the secondfinite-element model) and the condensed structural properties (containedin the first model with nine degrees of freedom), which is extremelyuseful in the construction of a tire.

[0061] The method consists in linking the design parameters of the tire,characteristics such as the mixture and the belt, to the structuralparameters, for instance the structural stiffness and camber stiffnessof the tire, because the quantities appearing in the model used have aphysical significance. This means that these quantities are directlylinked to the design parameters, in other words the model used is aphysical model. By so doing, any change to the design parameters of thetire leads to a change of the parameters of the predictive physicalmodel of the tire and this change, in turn, produces a variation of thetire's structural parameters.

[0062] The model permits identification of the structural parametersstarting from dynamic analysis made on the second, finite-element modelof the non-rolling tire. One requirement of the concentrated-parametermodel is, in fact, that it be predictive of actual behaviour of thetire.

[0063] One of the main advantages of the method according to theinvention is that the concentrated parameters are not identified bymeans of experimental tests on prototypes, but by means of virtualdynamic tests on the finite-element model of the non-rolling tire.

[0064] The method according to the invention uses a model of contactbetween tire and road that enables forecasting of the drift curves atsteady state. Also fed into the brush model, in addition to thelongitudinal and transversal stiffness of the inserts of the tread, wastheir torsional stiffness, these stiffnesses being identified by meansof numerical simulations on the second finite-element model, without anyneed for experimental tests.

[0065] The method according to the invention enables drift curves to bedetermined

[0066] by applying a drift angle α to the first physical model;

[0067] created in the area of contact on account of the drift are asideward force and a self-aligning torque that act on the first physicalmodel as forces acting on the hub and cause a lateral displacement and asnaking motion of the plate of the brush model;

[0068] because of the snaking motion σ_(b) and the lateral displacementof the plate, the forces set up in the contact area are modified; thisresults in a variation of the unconstrained degrees of freedom, amongwhich those of the plate, and therefore of the forces acting of thefirst physical model.

[0069] With this procedure, after a certain number of iterations, apoint is reached at which the degrees of freedom of the first physicalmodel settle about a steady state value. In this situation, the sidewardforce and the self-aligning torque created in the area of contact andfrom which the drift curves may be obtained are determined.

[0070] The method according to the invention also enables transientstate drift behaviour of the tire to be evaluated, making allowance inthe brush model for the dynamic deformations undergone by the inserts ofthe tread in this stage.

[0071] In this way, the length of relaxation of the tire while driftingis determined upon variation of the running conditions (speed, verticalload, drift angle, etc.). This procedure is also implemented without anyneed for experimental testing.

[0072] Characteristics and advantages of the invention will now bedescribed with reference to an embodiment of the invention, illustratedindicatively and by no means exclusively in the accompanying drawings,where:

[0073]FIG. 1 shows a concentrated-parameter physical model of a tireused in a method for determining the road handling of a tire of a wheelfor a vehicle, constructed according to the invention;

[0074]FIG. 2 shows a finite-element model of a tire used in the methodaccording to the invention;

[0075] FIGS. 3-6 are schematic representations of test procedures with anon-rolling tire to which the concentrated-parameter physical model ofFIG. 1 is subjected;

[0076] FIGS. 7-15 are graphs showing the results of the testsillustrated in the FIGS. 3-6, obtained from the finite-element model ofFIG. 2 which describes a selected real tire;

[0077] FIGS. 16-21 depict modes of vibration of the finite-element modeldescribing the selected real tire;

[0078]FIG. 22 is a flow diagram of a procedure for determining thesteady state drift curves of the concentrated-parameter physical modelof FIG. 1;

[0079]FIG. 23 is a schematic representation of a brush model associatedwith the concentrated-parameter physical model of FIG. 1;

[0080]FIG. 24 depicts a contact pressure distribution determined usingthe finite-element model describing the selected real tire;

[0081]FIG. 25 shows details of the brush model of FIG. 1;

[0082]FIGS. 26 and 27 are graphs illustrating the results obtained withthe brush model of FIG. 1;

[0083]FIG. 28 shows schematic representations of the brush model of FIG.1;

[0084]FIG. 29 depicts another contact pressure distribution determinedusing the finite-element model describing the selected real tire;

[0085]FIGS. 30 and 31 are drift curves obtained from theconcentrated-parameter physical model describing the selected real tire;

[0086]FIGS. 32 and 33 are schematic representations of drift transientstate test procedures to which the concentrated-parameter physical modelof FIG. 1 is subjected;

[0087]FIGS. 34, 35 and 36 are further schematic representations of thebrush model of FIG. 1;

[0088] FIGS. 37-52 illustrate the drift transient state test resultsobtained from the concentrated-parameter physical model describing theselected real tire; and

[0089]FIG. 53 depicts a distribution of forces acting on a beam of thetire brush model.

[0090] Illustrated in FIG. 1 is a concentrated-parameter physical model,indicated generically with the numeral 1, reproducing the driftbehaviour of a tire of a wheel, made from selected mixes of rubber andreinforcing materials.

[0091] The physical model 1 comprises a rigid ring 2 which represents atread band provided with inserts, a belting structure and correspondingcarcass portion of the tire, and a rigid disk 3 representing a hub ofthe wheel and beading of the tire. The model 1 also comprises principalsprings 4, 5 and 6 and principal dampers 7, 8 and 9 which connect therigid ring 2 to the hub 3 and represent sidewalls of the tire and airunder pressure inside the tire. The model also comprises supplementarysprings 10, 11 and 12 and supplementary dampers 13, 14 and 15 whichrepresent phenomena of deformation of the belt through the effect of aspecified vertical load.

[0092] Associated with the physical model 1 is a brush model 20 whichsimulates physical phenomena present in an area of contact between tireand road. The brush model 20 comprises a rigid plate 21 under which asystem representing the tread is applied. The system is preferablybidimensional and comprises numerous parallel, deformable beams 22orientated longitudinally, the ends of which are hinged to the plate,and numerous microinserts, or rows of springs, 23 arranged in parallel.In this particular case, there are three deformable beams 22 whilstthere are 5 rows of microinserts 23 associated with each beam. Thebottom ends of the microinserts 23 of the brush model interact with aroad or the ground 24. The model reproduces the local deformationsoccurring inside the area of contact and represents the uniformlydistributed lateral and torsional stiffnesses of the portion of tread inthe area of contact.

[0093] The rigid ring 2 has an equivalent roll radius r [m], mass M_(c)[kg] and diametral moment of inertia J_(c) [kg*m²]. The rigid disk 3 hasmass M_(m) [kg] and diametral moment of inertia J_(m) [kg*m²].

[0094] The principal springs 4, 5 and 6 have structural stiffnessesK_(c), respectively comprising lateral stiffness K_(cy) [N/m] betweenhub and belt, camber torsional stiffness K_(cθx) [Nm/rad] between huband belt and yawing torsional stiffness K_(cθz) [Nm/rad] between hub andbelt.

[0095] The principal dampers 7, 8 and 9 have structural dampings R_(c),respectively comprising lateral damping R_(cy) [Ns/m] between hub andbelt, camber torsional damping R_(cθx) [Nms/rad] between hub and beltand yawing torsional damping R_(cθz) [Nms/rad] between hub and belt.

[0096] The supplementary springs 10, 11 and 12 have residual stiffnessesK_(r), respectively comprising residual lateral stiffness K_(ry) [N/m],residual camber torsional stiffness K_(rθx) [Nm/rad] and residual yawingtorsional stiffness K_(rθz) [Nm/rad].

[0097] The supplementary dampers 13, 14 and 15 have residual dampingsR_(r), respectively comprising residual lateral damping R_(ry) [Ns/m],residual camber torsional damping R_(rθx) [Nms/rad] and residual yawingtorsional damping R_(rθz) [Nms/rad].

[0098] The residual stiffnesses and dampings permit allowance to be madefor the variation of local stiffness due to deflection of the tire. Thelateral and residual yawing stiffnesses K_(ry) and K_(rθx) connect thebottom end of the rigid ring to the plate, as also does the residualcamber stiffness K_(rθz). In some cases, the camber deformation of theplate ρ_(b) (absolute rolling rotation) is not taken into consideration,so that connecting the second end of the spring 11, which represents theresidual camber stiffness, directly to the plate is tantamount toconnecting it to the ground. In these cases, the stiffness K_(rθx)already incorporates the effect due to camber deformability of the brushmodel.

[0099] The equivalent system has stiffness per unit length c_(py) andthe contact area has a dynamic length 2a and dynamic width 2b.

[0100] With the equivalent system, allowance may be made both for thedeformability of the inserts in the tread and for the different speedsbetween a point of the insert in contact with the road (assuming thereis adhesion, this point has a lateral velocity y′_(s)=0) and thecorresponding point on the belt. Three factors play a decisive role: thecoefficient of friction at the interface between wheel and road, thenormal pressure distribution and stiffness of the inserts in the tread.

[0101] Shown in FIG. 1 is an absolute trio of reference axes O-X-Y-Zhaving versors i, j, k, where the origin O coincides with the centre ofthe hub with the tire non-deformed, the X axis lies in the plane of thehub and is of longitudinal direction, the Y axis is perpendicular to theX axis and the Z axis is vertical.

[0102] The degrees of freedom of the physical model 1 are:

[0103] absolute lateral displacement y_(m) of the hub, absolute yawrotation σ_(m) of the hub and absolute rolling rotation ρ_(m) of thehub,

[0104] relative lateral displacement y_(c) of the belt with respect tothe hub, relative yaw rotation σ_(c) of the belt with respect to the huband relative rolling rotation ρ_(c) of the belt with respect to the hub,

[0105] absolute lateral displacement y_(b) of the plate, absolute yawrotation σ_(b) of the plate and absolute rolling rotation ρ_(b) of theplate.

[0106] A further degree of freedom is:

[0107] absolute lateral displacement y_(s) of the bottom ends of themicroinserts.

[0108] This degree of freedom has the objective of reproducing thesideward forces created under the contact and which are linked to therelative displacements between the top and bottom ends of themicroinserts. In the case of perfect adhesion of the microinserts, withthe tire not drifting, y_(s)=0.

[0109] Motion of the physical model is described by assuming smalldisplacements and small rotations of the hub.

[0110] The tire is described using a Finite-Element Model (F.E.M.) 30depicted in FIG. 2. The finite-element model 30 comprises first elements(bricks or shells or multilayer composites) with a selected number ofnodes, having appropriately selected form functions, preferably of thefirst or second order and, even more preferably, linear, and secondelements suitable for describing the reinforcing materials. Each firstelement has a first stiffness matrix which is determined using aselected characterization of the mixes and a second supplementarystiffness matrix which is determined using a selected characterizationof the reinforcing materials.

[0111] More than ten different types of mixes are usually to be foundinside a tire. Their elastic properties are fed into the finite-elementmodel after selected static and dynamic tests are conducted on specimensof the mixes.

[0112] The static tests consist of tensile, compression and shear testsin which the test conditions, forces and elongations applied areestablished in relation to the properties of the mixes measuredpreviously (hardness, etc.).

[0113] The Mooney-Rivlin law of hyperelasticity was taken as theconstitutive law. This law describes the specific deformation energy inrelation to the derivatives of the displacements (deformations),separating the form variation energy from the volume variation energy(deviatoric and hydrostatic part of the stress tensor).

[0114] The coefficients of the constitutive law are calculated in such away as to minimize the difference between the experimental and thecalculated deformation energy.

[0115] The dynamic tests are conducted by applying first a staticpredeformation to the specimens and then an oscillating load withfrequency in the range from 0.1 to 100 Hz. The dynamic modulus of themixture is thus detected as the complex ratio between stress anddeformation. As the frequency is changed, the modulus and relative phasebetween stress and deformation are measured.

[0116] The reinforcing materials used are fabrics and of metallic type.In the tires for automobiles, metallic cords are used only for the beltsand fabric cords for the carcass and the outermost belt (zero degrees)located just under the tread band. The metallic cord is subjected topulling until it breaks and to compression in order to obtain byexperimental means the characteristic of the cord from critical load tobreaking. This characteristic is implemented in the finite-elementmodel. The fabric cord is also subjected to pulling until it breaks.

[0117] The second elements that describe the reinforcing materials aredefined within the first elements (bricks, for instance) of thefinite-element model 30, by assigning the geometrical disposition of thefabrics, the orientation of the cord, the spacing between the singlecords (thickness) and the experimentally obtained traction andcompression characteristic of the cord. These characteristics, inrelation to the dimensions of the brick element taken, are resumed in asupplementary stiffness matrix which is overlaid on the mixturestiffness matrix, enabling extraction of the cord tensions.

[0118] A frequency domain analysis is performed, wherein a linearizedresponse is determined to a harmonic excitation based on the singledegrees of freedom of the physical model. The response is obtained byresolving a matrix system, complete with mass, damping and stiffnessmatrices. Linearization of the matrices is performed at the end of thepreliminary static determination stages so that non-linear behaviour ofthe actual tire is implicitly taken into account.

[0119] By defining the isotropic linear viscoelasticity, the damping andstiffness matrices in relation to frequency may be built. It followsthat the relation between stresses and deformations is considerablyinfluenced by the elastic behaviour (a higher modulus of elasticitycorresponds in particular to higher frequencies) and the damping of themix.

[0120] In the case of FIG. 2, the finite-element model comprises:

[0121] 17,500 elements (16,000 defined by the user+1,500 generatedautonomously, needed for definition of the constraints of the contactbetween the tire and the rim it is fitted on and the road);

[0122] approx. 36,000 nodes (19,000 defined by the user+approx. 17,000generated autonomously, needed for resolution of the hydrostatic part ofthe stress tensor);

[0123] approx. 74,000 degrees of freedom (19,000 nodes×3 translationaldegrees of freedom for each node+17,000 degrees of freedom associatedwith the elements for resolution of the hydrostatic part of the stresstensor).

[0124] The equations of motion of the concentrated-parameter physicalmodel 1 of the non-rolling tire are obtained using the Lagrange method.

[0125] The unknown parameters of these equations are identified bycomparing the vibrational dynamic response determined using thefinite-element model with that obtained from the concentrated-parameterphysical model.

[0126] The above-described degrees of freedom (independent variables orgeneralized coordinates) of the physical model are contained in a vectorx organized as follows:

x={y _(m) ρ_(m) σ_(m) y _(c) ρ_(c) σ_(c) y _(b) ρ_(b) σ_(b)}^(T)   (1.1)

[0127] The kinetic energy, expressed through the independent variablesof the model, is as follows: $\begin{matrix}\begin{matrix}{E_{c} = \quad {{\frac{1}{2}M_{m}{\overset{.}{y}}_{m}^{2}} + {\frac{1}{2}{M_{c}\left( {{\overset{.}{y}}_{m} + {\overset{.}{y}}_{c}} \right)}^{2}} + {\frac{1}{2}J_{m}{\overset{.}{\rho}}_{m}^{2}} +}} \\{\quad {{\frac{1}{2}J_{m}{\overset{.}{\sigma}}_{m}^{2}} + {\frac{1}{2}{J_{c}\left( {{\overset{.}{\rho}}_{m} + {\overset{.}{\rho}}_{c}} \right)}^{2}} + {\frac{1}{2}{J_{c}\left( {{\overset{.}{\sigma}}_{m} + {\overset{.}{\sigma}}_{c}} \right)}^{2}}}}\end{matrix} & (1.2)\end{matrix}$

[0128] The potential energy, expressed through functions of theindependent variables of the model, is as follows: $\begin{matrix}\begin{matrix}{V = \quad {{{\frac{1}{2} \cdot K_{cy}}y_{c}^{2}} + {\frac{1}{2}K_{c\quad \theta \quad x}\rho_{c}^{2}} + {\frac{1}{2}K_{c\quad \theta \quad z}\sigma_{c}^{2}} + {\frac{1}{2}{K_{ry}\left( {y_{p} - y_{b}} \right)}^{2}} +}} \\{\quad {{\frac{1}{2}{K_{r\quad \theta \quad z}\left( {s_{p} - \sigma_{b}} \right)}^{2}} + {\frac{1}{2}{K_{r\quad \theta \quad x}\left( {r_{p} - \rho_{b}} \right)}^{2}}}}\end{matrix} & (1.3)\end{matrix}$

[0129] where

[0130] y_(p) is the absolute lateral displacement of the bottom point ofthe belt;

[0131] s_(p) is the absolute yaw rotation of the belt;

[0132] r_(p) is the absolute camber rotation of the belt.

[0133] The relations linking the physical variables to the independentvariables are as follows:

y _(p) =y _(m) +y _(c) +r*ρ _(m) +r*ρ _(c)

s _(p)=σ_(m)+σ_(c)

r _(p)=ρ_(m)+ρ_(c)

[0134] When these relations are inserted in the potential energyequation, the potential energy in relation to the generalizedcoordinates is obtained. The dissipation energy D is similar in form tothe potential energy V.

[0135] If Lagrange's theorem is applied to the kinetic energy expression(1.2), the inertia terms [M]* {umlaut over (x)} are found, where thegeneral mass matrix [M] is: $\begin{matrix}{\lbrack M\rbrack = \begin{bmatrix}{M_{m} + M_{c}} & 0 & 0 & M_{c} & 0 & 0 & 0 & 0 & 0 \\0 & {J_{m} + J_{c}} & 0 & 0 & J_{c} & 0 & 0 & 0 & 0 \\0 & 0 & {J_{m} + J_{c}} & 0 & 0 & J_{c} & 0 & 0 & 0 \\M_{c} & 0 & 0 & M_{c} & 0 & 0 & 0 & 0 & 0 \\0 & J_{c} & 0 & 0 & J_{c} & 0 & 0 & 0 & 0 \\0 & 0 & J_{c} & 0 & 0 & J_{c} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}} & (1.4)\end{matrix}$

[0136] The matrix includes the concentrated parameters of the modelM_(m), M_(c), J_(m) and J_(c).

[0137] On derivation of the expression of the potential energy V withrespect to the vector of the independent variables, a general stiffnessmatrix [K] is obtained (1.5): $\lbrack K\rbrack = \begin{bmatrix}K_{ry} & {K_{ry}r} & 0 & K_{ry} & {K_{ry}r} & 0 & {- K_{ry}} & 0 & 0 \\{K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x}} & 0 & {K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x}} & 0 & {{- K_{ry}}r} & {- K_{r\quad \theta \quad x}} & 0 \\0 & 0 & K_{r\quad \theta \quad z} & 0 & 0 & K_{r\quad \theta \quad z} & 0 & 0 & {- K_{r\quad \theta \quad z}} \\K_{ry} & {K_{ry}r} & 0 & {K_{cy} + K_{ry}} & {K_{ry}r} & 0 & {- K_{ry}} & 0 & 0 \\{K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x}} & 0 & {K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x} + K_{c\quad \theta \quad x}} & 0 & {{- K_{ry}}r} & {- K_{r\quad \theta \quad x}} & 0 \\0 & 0 & K_{r\quad \theta \quad z} & 0 & 0 & {K_{c\quad \theta \quad z} + K_{c\quad \theta \quad z}} & 0 & 0 & {- K_{r\quad \theta \quad z}} \\{- K_{ry}} & {{- K_{ry}}r} & 0 & {- K_{{ry}\quad}} & {{- K_{ry}}r} & 0 & K_{ry} & 0 & 0 \\0 & {- K_{r\quad \theta \quad x}} & 0 & 0 & {- K_{r\quad \theta \quad x}} & 0 & 0 & K_{r\quad \theta \quad x} & 0 \\0 & 0 & {- K_{r\quad \theta \quad z}} & 0 & 0 & {- K_{r\quad \theta \quad z}} & 0 & 0 & K_{r\quad \theta \quad z}\end{bmatrix}$

[0138] The structural and residual stiffnesses of the model are includedin this matrix.

[0139] Finally, on deriving the expression of the dissipation energy Dwith respect to the derivative vector before the independent variables,a general damping matrix [R] is obtained (1.6):$\lbrack R\rbrack = \begin{bmatrix}R_{ry} & {R_{ry}r} & 0 & R_{ry} & {R_{ry}r} & 0 & {- R_{ry}} & 0 & 0 \\{R_{ry}r} & {{R_{ry}r^{2}} + R_{r\quad \theta \quad x}} & 0 & {R_{ry}r} & {{R_{ry}r^{2}} + R_{r\quad \theta \quad x}} & 0 & {{- R_{ry}}r} & {- R_{r\quad \theta \quad x}} & 0 \\0 & 0 & R_{r\quad \theta \quad z} & 0 & 0 & R_{r\quad \theta \quad z} & 0 & 0 & {- R_{r\quad \theta \quad z}} \\R_{ry} & {R_{ry}r} & 0 & {R_{cy} + R_{ry}} & {R_{ry}r} & 0 & {- R_{ry}} & 0 & 0 \\{R_{ry}r} & {{R_{ry}r^{2}} + R_{r\quad \theta \quad x}} & 0 & {R_{ry}r} & {{R_{ry}r^{2}} + R_{r\quad \theta \quad x} + R_{c\quad \theta \quad x}} & 0 & {{- R_{ry}}r} & {- R_{r\quad \theta \quad x}} & 0 \\0 & 0 & R_{r\quad \theta \quad z} & 0 & 0 & {R_{c\quad \theta \quad z} + R_{c\quad \theta \quad z}} & 0 & 0 & {- R_{r\quad \theta \quad z}} \\{- R_{ry}} & {{- R_{ry}}r} & 0 & {- R_{{ry}\quad}} & {{- R_{ry}}r} & 0 & R_{ry} & 0 & 0 \\0 & {- R_{r\quad \theta \quad x}} & 0 & 0 & {- R_{r\quad \theta \quad x}} & 0 & 0 & R_{r\quad \theta \quad x} & 0 \\0 & 0 & {- R_{r\quad \theta \quad z}} & 0 & 0 & {- R_{r\quad \theta \quad z}} & 0 & 0 & R_{r\quad \theta \quad z}\end{bmatrix}$

[0140] The structural and residual dampings of the model are included inthis matrix.

[0141] The vector of the forces F contains the forces and torques actingon the physical model in the different situations the model is found in,as will be illustrated later.

[0142] In short, the equations of motion are obtained in matrix form:

[M].{umlaut over (x)}+[R].{dot over (x)}+[K].x=F   (1.7)

[0143] In this equation, the contributions linked respectively to thedegrees of freedom of the hub, the belt and the tread may be discerned:$\begin{matrix}{{{\begin{bmatrix}M_{m\quad m} & M_{m\quad c} & M_{mq} \\M_{c\quad m} & M_{cc} & M_{cq} \\M_{qm} & M_{qc} & M_{qq}\end{bmatrix} \cdot \overset{¨}{\underset{\_}{x}}} + {\begin{bmatrix}R_{m\quad m} & R_{m\quad c} & R_{mq} \\R_{c\quad m} & R_{cc} & R_{cq} \\R_{qm} & R_{qc} & R_{qq}\end{bmatrix} \cdot \underset{\_}{\overset{.}{x}}} + {\begin{bmatrix}K_{m\quad m} & K_{m\quad c} & K_{mq} \\K_{c\quad m} & K_{cc} & K_{cq} \\K_{qm} & K_{qc} & K_{qq}\end{bmatrix} \cdot \underset{\_}{x}}} = \begin{Bmatrix}F_{m} \\F_{c} \\F_{q}\end{Bmatrix}} & (1.8)\end{matrix}$

[0144] To identify the concentrated parameters of the physical model, aseries of virtual dynamic tests, simulated on the finite-element model30, is defined, in which the tire is variously excited and underdifferent conditions so as to highlight the contribution of each of theterms that has to be identified.

[0145] For each test, the finite-element model is used to determine thefrequency response of particular quantities measured at precise pointsof the tire.

[0146] To obtain the frequency responses of the concentrated-parameterphysical model, the equations of motion for the selected tests aredefined.

[0147] For each test and for each frequency response of the quantitiestaken into consideration (displacements of the points considered,constraining reactions, etc.), the difference is computed between theresults obtained from the finite-element model and those obtained usingthe concentrated-parameter physical model. This difference is consideredan error and is defined as

err={v−v _(FEM)}  (1.9)

[0148] An objective function is then also defined, consisting of theweighted sum of the differences found on each channel (each channelcorresponds to a quantity considered) between the responses of theconcentrated-parameter physical model and those of the finite-elementmodel. The objective function may be expressed as follows:$\begin{matrix}{f = {{\sum\limits_{i = 1}^{N\quad {channels}}{{v_{i} - v_{i\quad {FEM}}}}} + P_{i}}} & \text{(1.10)}\end{matrix}$

[0149] Finally each stiffness and damping parameter of the physicalmodel is attributed that value by means of which the objective functionis minimized, where the vector of the weightings P⁻ enables the variouschannels to be given different weightings.

[0150] To identify the values of the three stiffnesses and the threedampings present between the rigid ring and the hub, two virtual testsare conducted on the tire blown up and not pressed to the ground. Alsoobtained at this stage of the identification are the values for the massand diametral moment of inertia of the belt that most closely adaptbehaviour of the model to that of the actual tire. In particular, thevalue of the mass of the belt M_(c) is determined in the first test andthe value of the diametral moment of inertia of the belt J_(c) in thesecond test.

[0151] The mass of the hub M_(m), on the other hand, is determined byimposing the conservation of the total mass of the tire:M_(tot)=M_(c)+M_(m). A first sharing of the masses is then made with aview to defining the first-attempt values of the mass of the belt andits diametral moment of inertia. These values will then be modified inthe identification stage in order to optimize dynamic behaviour of themodel.

[0152] A first test (test A) on the finite-element model consists inexciting the blown-up tire not pressed to the ground by imposing atranslation in the Y direction on the hub. The lateral displacement atparticular points of the belt of the tire and the force set up betweenhub and belt are measured. A schematic representation of the first testis shown in FIG. 3 depicting belt 32, hub 33 in transverse section and aconnection spring 34, which represents the lateral stiffness and dampingbetween belt and hub. Also shown are an arrow 31 representing thedisplacement imposed on the hub in the Y direction, and a side view ofthe belt 32, in which the cardinal points N, E, S and W are shown. Asthe belt is symmetrical, the lateral displacement of one cardinal pointonly of the belt 32 is considered in this test.

[0153] A first comparison test (A) is carried out on theconcentrated-parameter tire model, as will be illustrated later. Whenthe frequency responses of the quantities measured in these first twotests (lateral displacement of a cardinal point of the belt and forcebetween belt and hub) are compared and the errors minimized, thefollowing concentrated parameters of the model 1 may then be determined:

[0154] the lateral stiffness between hub and belt K_(cy),

[0155] the lateral damping between hub and belt R_(cy),

[0156] the mass of the belt M_(c).

[0157] A second test (test B) on the finite-element model consists inexciting the blown-up tire not pressed to the ground by imposing acamber rotation on the hub. The lateral displacement at particularpoints of the belt (camber rotation) and the torque transmitted betweenhub and belt are measured. A schematic representation of the second testis shown in FIG. 4 depicting the belt 32, the hub 33 in transversesection and a connection spring 35, which schematically represents thecamber torsional stiffness and damping between belt and hub,respectively K_(cθx) and R_(cθx). Also shown are an arrow 41representing the rotation imposed on the hub about the X axis, and aside view of the belt 32, in which the cardinal points N, E, S and W areindicated. As the belt is symmetrical, the lateral displacement of onecardinal point only of the belt 32, selected indifferently betweenpoints N and S, is considered in this test.

[0158] A second comparison test (B) is carried out on theconcentrated-parameter tire model, as will be illustrated later. Againin this case, when the frequency responses of the quantities measured inthese second two tests (lateral displacement of a cardinal point of thebelt and torque between belt and hub) are compared and the errorsminimized, the following concentrated parameters of the model 1 may thenbe determined:

[0159] the diametral moment of inertia of the belt J_(c),

[0160] the camber torsional stiffness between hub and belt K_(cθx),

[0161] the camber torsional damping between hub and belt R_(cθx),

[0162] the yawing torsional stiffness between hub and belt K_(cθz) (bysymmetry with K_(cθx))

[0163] the yawing torsional damping between hub and belt R_(cθz) (bysymmetry with R_(cθx)).

[0164] The connection between hub and belt is made in like manner forboth the camber rotations and the yaw rotations and it is possible toconsider that the behaviour of the belt not pressed to the ground isaxial-symmetrical.

[0165] A third test (test C) on the finite-element model consists inexciting the tire blown up, pressed to the ground and bereft of itstread (rasped) at least in the area of contact by the application to thehub of a force in the Y direction. The lateral displacement of the huband particular points of the belt are measured. A schematicrepresentation of the third test is shown in FIG. 5 depicting the belt32, the hub 33 in transverse section, a connection spring 36 (inaddition to the one described above, indicated with numeral 35) whichschematically represents the residual, lateral and camber stiffnessesand dampings between belt and ground 24. Also shown are an arrow 51representing the force imparted, an arrow 52 representing the lateraldisplacement of the hub and a front view of the belt 32, in which thecardinal points N, E, S and W are indicated. In this test, the lateraldisplacement of the N and E cardinal points of the belt 32 isconsidered.

[0166] A third comparison test (C) is carried out on theconcentrated-parameter tire model, as will be illustrated later. Whenthe frequency responses of the quantities measured in these third twotests (lateral displacement of the hub and of the N and E cardinalpoints of the belt) are compared and the errors minimized, the followingconcentrated parameters of the model 1 may then be determined:

[0167] the residual lateral stiffness K_(ry),

[0168] the residual lateral damping R_(ry),

[0169] the residual camber stiffness K_(rθx),

[0170] the residual camber damping R_(rθx).

[0171] A fourth test (test D) on the finite-element model consists inexciting the tire blown up, pressed to the ground and bereft of itstread (rasped) at least in the area of contact by the application to thehub of a yawing torque about the Z axis. The yaw rotation of the hub andthe displacement at particular points of the belt are measured. Aschematic representation of the fourth test is shown in FIG. 6 depictingthe belt 32, the hub 33 in transverse section, a connection spring 37(in addition to the one described above, indicated with numeral 35)which schematically represents the residual yawing stiffnesses anddampings between belt and ground 24. Also shown are an arrow 61representing the yawing torque imparted, an arrow 62 representing therotation of the hub and a front view of the belt 32, in which thecardinal points N, E, S and W are indicated. In this test, the lateraldisplacement of the E and W cardinal points of the belt 32 isconsidered.

[0172] A fourth comparison test (D) is carried out on theconcentrated-parameter tire model, as will be illustrated later. Whenthe frequency responses of the quantities measured in these fourth twotests (yaw rotation of the hub and displacement of the E and W cardinalpoints of the belt) are compared and the errors minimized, the followingconcentrated parameters of the model 1 may then be determined:

[0173] the residual yawing stiffness K_(rθz),

[0174] the residual yawing damping R_(rθz),

[0175] During the third and fourth tests, in addition to theidentification of the residual quantities, the stiffness and dampingvalues already identified in the first and second tests are alsomodified in order to better describe the transfer functions given by thefinite-element model. The stiffness and damping values between hub andbelt are therefore modified slightly from those identified earlier.

[0176] To obtain the frequency responses of the physical model 1 neededfor the identification of its concentrated parameters, the equations ofmotion relative to the four tests described above are obtained.

[0177] The tests A and B are carried out with the tire blown up and notpressed to the ground, imposing displacements on the hub.

[0178] The free degrees of freedom (generalized coordinates) of thephysical model are:

x_(i) ^(A−B) ={y _(c) ρ_(c) σ_(c) y _(b) ρ_(b) σ_(b)}^(T)   (1.11)

[0179] whereas the constrained degrees of freedom (constrainedcoordinates) are:

x_(u) ^(A−B) ={y _(m) ρ_(m) σ_(m)}^(T)   (1.12)

[0180] As stated above, only a translation in the Y direction is imposedon the hub in test A and therefore ρ_(m)=σ_(m)=0, whilst only a camberrotation is imposed on the hub in test B and therefore y_(m)=σ_(m)=0.Furthermore the tire is raised off the ground and the residualstiffnesses and dampings relative to the tread cancel each other out.

[0181] Under these conditions, from the general matrices of mass,stiffness and damping reported above for the concentrated-parameterphysical model, a mass matrix [M]^(A−B), a stiffness matrix [K]^(A−B)and a damping matrix [R]^(A−B) are obtained.

[0182] When a partition is made of the matrices in relation to the freeand constrained coordinates, equations of motion in scalar form areobtained from (1.8): $\begin{matrix}\left\{ \begin{matrix}{{{\left( {M_{m} + M_{c}} \right){\overset{¨}{y}}_{m}} + {M_{c}\overset{¨}{y_{c}}}} = F_{ym}} \\{{{\left( {J_{m} + J_{c}} \right){\overset{¨}{\rho}}_{m}} + {J_{c}{\overset{¨}{\rho}}_{c}}} = C_{\rho \quad m}} \\{{{\left( {J_{m} + J_{c}} \right){\overset{¨}{\sigma}}_{m}} + {J_{c}{\overset{¨}{\sigma}}_{c}}} = C_{\sigma \quad m}}\end{matrix} \right. & (1.13)\end{matrix}$

[0183] where F_(ym), C_(ρm) and C_(σm) represent the forces and torquesimposed on the hub.

[0184] These equations govern the motion of the system in the simulatedtests A and B.

[0185] For the displacements imparted to the hub, the equations are:$\begin{matrix}\left\{ \begin{matrix}{{{M_{c}{\overset{¨}{y}}_{c}} + {R_{cy}{\overset{.}{y}}_{c}} + {K_{cy}y_{c}}} = {{- M_{c}}{\overset{¨}{y}}_{m}}} \\{{{J_{c}{\overset{¨}{\rho}}_{c}} + {R_{c\quad \theta \quad x}{\overset{.}{\rho}}_{c}} + {K_{c\quad \theta \quad x}\rho_{c}}} = {{- J_{c}}{\overset{¨}{\rho}}_{m}}} \\{{{\left( {J_{m} + J_{c}} \right){\overset{¨}{\sigma}}_{m}} + {J_{c}{\overset{¨}{\sigma}}_{c}}} = C_{\sigma \quad m}}\end{matrix} \right. & (1.14)\end{matrix}$

[0186] The frequency responses must now be determined of the quantitiesmeasured in the tests A and B to be able to compare the frequencyresponses given by the finite-element model with the frequency responsesof the concentrated-parameter model.

[0187] In the test A, with the physical model, the frequency response ofthe lateral displacement of the belt is reconstructed, taking intoconsideration the first equation of the system (1.14). The displacementimparted on the hub is:

y _(m) ^(A) =A.e ^(iΩt)   (1.15)

[0188] the complete differential equation to be resolved is as follows:

M _(c) ÿ _(c) ^(A) +R _(cy) {dot over (y)} _(c) ^(A) +K _(cy) y _(c)^(A) =M _(c).Ω² .A.e ^(iΩt)   (1.16)

[0189] The solution of this equation, having a sinusoidal force agentequivalent to y_(m), is of the type:

y _(c) ^(A) =B.e ^(lΩt)   (1.17)

[0190] By means of substitutions and simplifications, the frequencyresponse is obtained for the degree of freedom y_(c) upon variation ofΩ, that is to say the frequency response of the lateral displacement ofthe belt with respect to the hub. $\begin{matrix}{y_{c}^{A} = {\frac{M_{c} \cdot \Omega^{2} \cdot A}{\left( {{{{- M_{c}} \cdot O^{2}} + i}{{\cdot \Omega \cdot R_{cy}} + K_{cy}}} \right)} \cdot ^{\quad \Omega \quad t}}} & (1.18)\end{matrix}$

[0191] As the finite-element model provides the absolute displacement ofthe points of the belt, the frequency response of the absolute lateraldisplacement of the belt may be determined from the absolute lateraldisplacement: $\begin{matrix}\begin{matrix}{y_{c\quad \_ \quad {ass}}^{A} = {y_{c}^{A} + y_{m}^{A}}} \\{= {\left\lbrack {\frac{M_{c} \cdot \Omega^{2}}{\left( {{{- M_{c}} \cdot \Omega^{2}} + {i \cdot \Omega \cdot R_{cy}} + K_{cy}} \right)} + 1} \right\rbrack {A \cdot ^{\quad \Omega \quad t}}}}\end{matrix} & (1.19)\end{matrix}$

[0192] Upon computing the difference between the lateral displacement ofthe physical model and that provided by the finite-element model, anerror is obtained that is a function of M_(c), K_(cy) and R_(cy). In theidentification stage, this error is minimized, that is to say, valuesare chosen for M_(c), K_(cy) and R_(cy) that make the error minimum.

[0193] In the test A, the frequency response of the force transmittedbetween hub and belt provided by the finite-element model is alsocompared with that provided by the concentrated-parameter physicalmodel. This force is equal to:

F _(hub-belt) ^(A)=(K _(cy) +i.Ω.R _(cy)).B.e ^(iΩt)   (1.20)

[0194] Again with this quantity, when the difference is computed betweenthe frequency response of the force determined with the finite-elementmodel and that obtained using the concentrated-parameter physical model,a second error is obtained that is also function of M_(c), K_(cy) andR_(cy). The above-stated quantities are obtained upon minimization ofthis error.

[0195] In the test B, with the physical model, the frequency response ofthe rotation of the belt is reconstructed and the same method ofprocedure as the test A is then followed. The second equation of thesystem (1.14) is considered, starting from a camber rotation imparted tothe hub of the type:

ρ_(m) ^(B) =C.e ^(iΩt)   (1.21)

[0196] the frequency response is obtained of the absolute camberrotation of the belt: $\begin{matrix}\begin{matrix}{\rho_{c\quad \_ \quad {ass}}^{B} = {\rho_{c}^{B} + \rho_{m}^{B}}} \\{= {\left\lbrack {\frac{J_{c} \cdot \Omega^{2}}{\left( {{{- J_{c}} \cdot \Omega^{2}} + {i \cdot \Omega \cdot R_{c\quad \theta \quad x}} + K_{c\quad \theta \quad x}} \right)} + 1} \right\rbrack \cdot C \cdot ^{i\quad \Omega \quad t}}}\end{matrix} & (1.22)\end{matrix}$

[0197] and, from this, the frequency response of the rotation of thelateral displacement of the S point of the belt is determined:

y _(c) _(—) _(ass-south) ^(B)=ρ_(c) _(—) _(ass) ^(B) .r   (1.23)

[0198] Also determined is the frequency response of the torquetransmitted between hub and belt:

C _(hub-belt) ^(B)=(K _(cθx) +i.Ω.R _(cθx)).D.e ^(lΩt)   (1.24)

[0199] Upon computing the difference between the displacement y_(c) _(—)_(ass) _(—) _(south) ^(B) and the torque C_(hub-belt) of the physicalmodel and those given by the finite-element model, an error is obtainedthat is a function of the quantities J_(c), K_(cθx) and R_(cθx). In theidentification stage, this error is minimized and values are identifiedfor the above-mentioned quantities.

[0200] In the tests C and D, with the tire blown up, pressed to theground and bereft of tread, the free degrees of freedom and theconstrained degrees of freedom of the physical model are determined inthe two test situations and an axial force is imposed on the hub in theY direction, in the test C, and a yawing torque about the Z axis, in thetest D. When these conditions are put into the equations of motion (1.7)and (1.8), the equations of motion for the free degrees of freedom ofthe physical model are obtained and the complex vector of the freedegrees of freedom is determined. This vector contains both the modulusand the phase of the free degrees of freedom in question.

[0201] In the test C, from this vector, the frequency responses of thelateral displacement of the hub, of the N point and of the E point ofthe belt are determined. In the identification stage, the difference iscomputed between the frequency response obtained from the finite-elementmodel and that obtained from the concentrated-parameter physical modeland a second error is obtained which is a function of K_(ry), R_(ry),K_(rθx) and R_(rθx). On minimizing this error, the above-mentionedquantities are determined.

[0202] In the test D, from the above-mentioned vector, the frequencyresponses are determined of the yaw rotation of the hub and of thedisplacement of the E point of the belt in relation to the quantitiesK_(rθ) _(z) and R_(rθz). These quantities are determined by minimizingthe error resulting as the difference between the frequency responseprovided by the finite-element model and that provided by theconcentrated-parameter physical model.

[0203] Once the concentrated parameters of the physical model have beenidentified, it is possible to determine some quantities that allow anevaluation, or an initial approximation at least, to be made of thebehaviour of the tire. These quantities are the total drift stiffness ofthe tire K_(d), comprised in turn by the structural stiffness K_(c), thetread stiffness K_(b) and the total camber stiffness K_(γ) of the saidtire.

[0204] The stiffness K_(d) is an important parameter for the purposes ofdefinition of the tire in that it provides an indication of the effectthat the design parameters have on the drift behaviour.

[0205] To obtain K_(d), the free and constrained degrees of freedom arepartitioned:

x _(l) ={y _(c) ρ_(c) σ_(c) y _(b) σ_(b)}  (1.25)

[0206] A further partition is made of the vector x _(l) into internal(of the belt) and external (of the tread) degrees of freedom:

x _(l) ={x _(e) x _(i)}^(T) ={{y _(b) σ_(b) } {y _(c) ρ_(c) σ_(c)}}^(T)  (1.26)

[0207] The stiffness matrix (1.5) is then modified following thesepartitions and the equations of motion can be written as follows for thefree degrees of liberty: $\begin{matrix}\left\{ \begin{matrix}{{{\left\lbrack K_{ee} \right\rbrack \cdot {\underset{\_}{x}}_{e}} + {\left\lbrack K_{ei} \right\rbrack \cdot {\underset{\_}{x}}_{i}}} = {\underset{\_}{F}}_{e}} \\{{{\left\lbrack K_{ie} \right\rbrack \cdot {\underset{\_}{x}}_{e}} + {\left\lbrack K_{ii} \right\rbrack \cdot x_{i}}} = {{\underset{\_}{F}}_{i} = \underset{\_}{0}}}\end{matrix} \right. & (1.27)\end{matrix}$

[0208] Remembering that no external forces act on the degrees ofinternal freedom, the second equation of the system (1.27) may beresolved in relation to the external degrees of freedom, the resultinserted in the first equation and the latter made explicit in relationto the external coordinates x_(e) alone. At this point a stiffnessmatrix is defined for the tire structure, limited to the externaldegrees of freedom: $\begin{matrix}\left\{ \begin{matrix}{{{\left\lbrack K_{ee} \right\rbrack \cdot {\underset{\_}{x}}_{e}} - {{\left\lbrack K_{ei} \right\rbrack \cdot {\left\lbrack K_{ii} \right\rbrack^{- 1}\left\lbrack K_{ie} \right\rbrack}}{\underset{\_}{x}}_{e}}} = {\underset{\_}{F}}_{e}} \\{{\underset{\_}{x}}_{i} = {{- {\left\lbrack K_{ii} \right\rbrack^{- 1}\left\lbrack K_{ie} \right\rbrack}} \cdot x_{e}}}\end{matrix} \right. & (1.28)\end{matrix}$

[0209] The resolving equation is:

[{circumflex over (K)}_(ee)]x _(e)=F _(e)   (1.29)

[0210] wherein:

[{circumflex over (K)} _(ee) ]=[K _(ee) ]−[K _(ei) ][K _(ii)]⁻¹ .[K_(ie)]  (1.30)

[0211] is a matrix (2,2). The first element of the first row of thismatrix represents the total stiffness of the tire K_(d):

K_(d)={circumflex over (K)}_(ee)(1,1)   (1.31)

[0212] With the stiffnesses of the tire with concentrated parametersknown from the identification procedure, it is possible to calculateK_(d).

[0213] The total camber stiffness K_(γ) of the tire provides anindication of the tire's ability to exit from a longitudinal track,namely parallel to the direction of movement of the tire, made on theroad when the tire is moving with rectilinear motion. The methodaccording to the invention enables calculation of K_(γ) without the needto make and test a tire prototype, as is generally done.

[0214] To determine K_(γ), a virtual test, practically equal to theexperimental one, is carried out consisting in imposing a selectedcamber angle on the hub, with a null drift angle, and in measuring thesideward force created on the hub.

[0215] The free and constrained degrees of freedom are identified:

x _(l) ^(γ) ={y _(c) ρ_(c) σ_(c) y _(b) σ_(b)}  (1.32)

x _(v) ^(γ) ={y _(m) ρ_(m) σ_(m) ρ_(b)}  (1.33)

[0216] By imposing a camber angle and leaving all the other constraineddegrees of freedom unaltered, the sideward force is determined byresolving the matrix equation:

[K]x=F  (1.34)

[0217] where the matrix [K] is the (1.5) and F a vector containing theforces acting on the tire. Figuring among these are the force and torquedue to the deformations undergone by the inserts of the tread in theassumption that there is perfect adhesion. These two contributions are:$\begin{matrix}{{F_{yb} = {2 \cdot c_{p} \cdot a \cdot y_{b}}}{M_{zb} = {\frac{2}{3} \cdot c_{p} \cdot a^{3} \cdot \sigma_{b}}}} & (1.35)\end{matrix}$

[0218] As the previous two terms are functions of two of the ninedegrees of freedom of the concentrated-parameter model (y_(b) andσ_(b)), they must be expressed in function of these degrees of freedomand thus be brought to the left-hand side of the equals sign in thematrix equation reported above (1.34), so that they too contribute todetermining the stiffness matrix of the system.

[0219] The partitioned matrices K_(ll) ^(γ), K_(lv) ^(γ), K_(vl) ^(γ)and K_(vv) ^(γ) are known: $\begin{matrix}{\left\lbrack K_{vv} \right\rbrack^{y} = \begin{bmatrix}K_{ry} & {K_{ry}r} & 0 & 0 \\{K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x}} & 0 & {- K_{r\quad \theta \quad x}} \\0 & 0 & K_{r\quad \theta \quad z} & 0 \\0 & {- K_{r\quad \theta \quad x}} & 0 & K_{r\quad \theta \quad x}\end{bmatrix}} & (1.36) \\{\left\lbrack K_{vi} \right\rbrack^{y} = \begin{bmatrix}K_{ry} & {K_{ry}r} & 0 & K_{ry} & 0 \\{K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x}} & 0 & {{- K_{ry}}r} & 0 \\0 & 0 & K_{r\quad \theta \quad z} & 0 & {- K_{r\quad \theta \quad z}} \\0 & {- K_{r\quad \theta \quad x}} & 0 & 0 & 0\end{bmatrix}} & (1.37) \\{\left\lbrack K_{lv} \right\rbrack^{y} = \begin{bmatrix}K_{ry} & {K_{ry}r} & 0 & 0 \\{K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x}} & 0 & {- K_{r\quad \theta \quad x}} \\0 & 0 & K_{r\quad \theta \quad z} & 0 \\{- K_{r\quad y}} & {{- K_{{ry}\quad}}r} & 0 & 0 \\0 & 0 & {- K_{r\quad \theta \quad z}} & 0\end{bmatrix}} & (1.38) \\{\left\lbrack K_{ll} \right\rbrack^{y} = \begin{bmatrix}{K_{cy} + K_{ry}} & {K_{ry}r} & 0 & {- K_{ry}} & 0 \\{K_{ry}r} & {{K_{ry}r^{2}} + K_{r\quad \theta \quad x} + K_{c\quad \theta \quad x}} & 0 & {{- K_{ry}}r} & 0 \\0 & 0 & {K_{c\quad \theta \quad z} + K_{r\quad \theta \quad z}} & 0 & {- K_{r\quad \theta \quad z}} \\{- K_{r\quad y}} & {{- K_{{ry}\quad}}r} & 0 & {K_{ry} - {2 \cdot c_{p} \cdot a}} & 0 \\0 & 0 & {- K_{r\quad \theta \quad z}} & 0 & {K_{r\quad \theta \quad z} - {\frac{2}{3} \cdot c_{p} \cdot a^{3}}}\end{bmatrix}} & (1.39)\end{matrix}$

[0220] From these, the vector of the forces acting on the constraineddegrees of freedom may be calculated using the equation:

F _(v) ^(γ) ={−[K _(vl)]^(γ) .[K _(ll)]^(γ−1) .[K _(lv)]^(λ) +[K_(vv)]^(γ) }x _(v) ^(γ)  (1.40)

[0221] Remembering that:

[{circumflex over (K)}] ^(γ) =−[K _(vl)]^(γ) .[K _(ll)]^(γ−1) .[K_(iv)]^(λ) +[K _(vv)]^(γ)  (1.41)

[0222] is a matrix (4,4), the first element of the second row, which isthe total camber stiffness of the tire, is determined:

K _(γ) ={circumflex over (K)} ^(γ)(1, 2)   (1.42)

[0223] To demonstrate the validity of the method according to theinvention, the results obtained are brought into the procedure for theidentification of the concentrated parameters of the physical model of atire of the 55 range (H/C section ratio of 0.55) manufactured by theApplicant. In the graphs of FIGS. 7-15, the frequency response obtainedwith the concentrated-parameter physical model is represented with acontinuous line, while the frequency response obtained with thefinite-element model during the tests A, B, C and D is represented withasterisks.

[0224]FIG. 7 is a graph of the frequency response of the displacement ofa point of the belt obtained during the test A from the physical modeland from the finite-element model, whereas FIG. 8 is a graph of thefrequency response of the force created between the hub and the belt,again in the test A.

[0225] The values of the concentrated parameters identified with thetest A are reported in the following Table I. TABLE I Parameter SymbolValue identified Linear stiffness between hub and belt⁹ K_(cy) 5 e⁵[N/m] Linear damping between hub and belt R_(cy) 116.906 [Ns/m] Beltmass M_(c) 7.792 [kg]

[0226] It may be observed from the graphs of FIGS. 7 and 8 that thefrequency response has a single resonance peak. Corresponding to thispeak is a mode of vibration of the tire illustrated in FIG. 16. Thismode of vibration maintains the tread band substantially rigid and maytherefore be described by the concentrated-parameter model, thatsimulates the sidewall and the belt with a rigid ring. The parametersidentified are therefore valid in a set of frequencies ranging from 0 to100 Hz, since modes of vibration that considerably deform both sidewalland belt appear with frequencies that are higher than this.

[0227]FIG. 9 is a graph of the frequency response of the displacement ofa point of the belt obtained during the test B from the physical modeland from the finite-element model, whereas FIG. 10 is a graph of thefrequency response of the torque created between the hub and the belt,again in the test B.

[0228] The values of the concentrated parameters identified with thetest B are reported in the following Table II. TABLE II Parameter SymbolValue identified Torsional stiffness between hub and beltK_(cex)/K_(cez) 3.3 e⁴ [N/m] Torsional damping between hub and beltR_(cex)/R_(cez) 8.217 [Ns/m] Diametral moment of inertia of belt J_(c)0.373 [kg*m²]

[0229] As in the previous test, again in this one there is a singleresonance peak corresponding to the modal deformation of the tire shownin FIG. 17.

[0230] In the test C, consideration also needs to be given to the normalload bearing on the tire. Three loads corresponding to three standardworking conditions were applied to the tire: a reduced load of between2,500 and 3,000 N, an intermediate load of between 3,500 and 4,800 N anda high load of between 5,100 and 6,500 N.

[0231]FIG. 11 is a graph of the frequency response of the lateraldisplacement of the hub obtained from the physical model and from thefinite-element model during the test C, with a normal load of 2,914 Nbearing on the tire, whereas FIGS. 12 and 13 are graphs of the frequencyresponse of the lateral displacements of the E and N points of the belt,again in the test C.

[0232] The values of the concentrated parameters identified in the testC are reported in the following Table III. TABLE III Parameter SymbolValue identified Residual linear stiffness K_(ry) 665964 [N/m] Residualcamber stiffness R_(r0x) 11461 [Nm/rad] Residual linear damping R_(ry)2042.345 [Ns/m] Residual camber damping R_(r0x) 0.358 [Nms/rad]

[0233] Unlike in the two earlier tests, there are two resonance peaks inthis third test corresponding to two modes of vibration of the tiredepicted in FIGS. 18 and 19.

[0234] Again in this case, the modes of vibration maintain the sidewalland belt complex substantially rigid and may therefore be describedaccurately by the concentrated-parameter model, in other words theconcentrated-parameter model is valid in the range of frequenciesbetween 0 and 100 Hz.

[0235] The frequency responses obtained with the loads of 4,611 N and6,302 N are not shown herein. The results, however, are similar to thoseshown for the 2,914 N load.

[0236]FIG. 14 is a graph of the frequency response of the yaw rotationof the hub obtained from the physical model and from the finite-elementmodel during the test D, with a normal load of 2,914 N bearing on thetire, whereas FIG. 15 is a graph of the frequency response of thelateral displacement of the E point of the belt, again in the test D.

[0237] The values of the concentrated parameters identified with thetest D are reported in the following Table IV. TABLE IV Parameter SymbolValue identified Residual yaw stiffness K_(r0z) 14933 [Nm/rad] Residualyaw damping R_(r0z) 14.318 [Nms/rad]

[0238] Again in the test D, there are two resonance peaks whichcorrespond to the two modes of vibration of the tire shown in FIGS. 20and 21.

[0239] In this case, whereas the first mode of vibration (FIG. 20) iscomparable to a rigid mode, it is much more difficult to describe thesecond one (FIG. 21) as a rigid mode. It seems apparent that the belt asa whole becomes deformed in this second mode of vibrating. Thereconstructed frequency response does not therefore accurately reproducethe second resonance peak. The concentrated-parameter model in the testD is therefore only valid in the range of frequencies from 0 to 70 Hz.

[0240] Again in the test D, the frequency responses obtained with theloads of 4,611 N and 6,302 N are not shown herein. The results, however,are similar to those shown for the 2,914 N load.

[0241] To evaluate a tire in relation to its road handling, verificationneeds to be made that the total drift stiffness K_(d) of the tire andthe total camber stiffness K_(γ) are within the following value ranges:

[0242] K_(d)=500-2,000 [N/g]

[0243] K_(γ)=40-3,500 [N/g]

[0244] K_(c)=8,000-30,000 [N/g]

[0245] K_(b)=150-400 [/g]

[0246] where g=degree.

[0247] The method according to the invention enables an evaluation to bemade of the steady state behaviour of the drifting tire. Thebidirectional brush model shown in FIG. 1 and illustrated above is used.

[0248] Under drift conditions, the microinserts of the brush modelbecome deformed and a sideward force and a moment of torque act on thebeam that they are connected to. These forces and moments result in adeformation of the beam that affects the configuration of themicroinserts. By means of successive iterations, the deformationeffectively assumed by the beams in relation to the drift angle imposedon the hub is determined. At this point, the total sideward force isdetermined, together with the total self-aligning torque acting on theplate that the beams are connected to. Under the total sideward forceand self-aligning torque, the plate snakes by an angle that depends onthe overlying structure, i.e. on the springs connecting it to the hub.On account of this snaking, the deformation of the microinserts isaltered. By performing new iterations, the sideward force andself-aligning torque that the tire summons up in reaction to the driftangle imposed are determined.

[0249] The procedure is illustrated in the flow diagram of FIG. 22.

[0250] A drift angle α is applied to the physical model 1, starting froma condition in which the beams 22 are in a non-deformed configurationand the brush model has a null snaking σ_(b) (block 45). The sidewardforces acting on the beams through the effect of the drift angle α anddepending on the difference α−σ_(b) and on the deformation of the beamsare determined (blocks 46 a, 46 b and 46 c). The deformation of thebeams is determined (blocks 47 a, 47 b and 47 c). A check is made to seeif the deformation is the same as that determined in the previous step(blocks 48 a, 48 b and 48 c). The procedure for determining deformationof the beams is repeated until deformation is verified to be equal tothat found in the previous step. At this point, the snaking of theplate, i.e. of the brush model, is determined (block 49). A check ismade to see if the snaking is equal to that calculated in the previousstep (block 50). The procedure for determining snaking of the plate isrepeated until snaking is verified to be equal to that found in theprevious step. At this point, the sideward force and the self-aligningtorque acting on the hub due to the drift angle imposed are determined(block 51). The procedure is repeated for the different values of thedrift angle α to produce drift, force and self-aligning torque curvesthat enable steady state drift behaviour of the tire to be evaluated.

[0251] The beams of the brush model do not all become deformed in thesame way since the pressure distribution acting on each beam isdifferent, as also are the sideward forces. In practice, the sidewallsof the tire (shoulders) have greater stiffness than in the central bandof the tread.

[0252] To identify the flexural stiffness of each beam, a distributedparameters model is used based on a model depicted in FIG. 23 whereinthe equivalent beams and springs (microinserts) of the brush model areindicated with the same numerals as in FIG. 1. A linear static analysis,corresponding to a sideways traction and made by applying a sidewardforce to the contact surface of the road, is then conducted on thefinite-element model. The accordingly stressed tire becomes deformed andthe lateral displacements are determined at circumferential sectionsthereof, on a level with the external band. These circumferentialsections are divided into three groups corresponding to the central partof the area of contact and to the two parts at either side. The meanlateral displacement of each group is calculated over the full length ofthe contact area. On determining the difference between the genericdeformation mean (mean lateral displacements over the full length of thecontact area) provided by the finite-element model and the lateraldeformation of the rigid ring of the concentrated-parameter physicalmodel, also subjected to the same sideward force as applied to thefinite-element model, a “difference” deformation is obtained that mustbe offset by the equivalent beam of flexural stiffness EJ (N*m²).

[0253] Knowing how the sideward force is divided over the three parts ofthe contact area from the linear static analysis of sideways traction onthe finite-element model, a flexural stiffness EJ is determined for eachbeam.

[0254] In FIG. 23 it may be seen that the sections closest to thesidewall that the traction is exerted on become more deformed than thosefurther away (dashed lines). If a load of F/3 were to be applied to eachbeam, there would be a monotone pattern of EJ when moving from one sideof the tire to the other, but this would be contrary to what experienceshows. Therefore, using the finite-element model, a horizontal divisionof the load between the various sections is also made.

[0255] The results of the identification procedure are reported in thefollowing Table V. TABLE V EJ [N*m²] 1^(st) load 2^(nd) load 3^(rd) load1^(st) beam 50 120 250 2^(nd) beam 45 106 210 3^(rd) beam 50 120 250

[0256] To describe as accurately as possible what happens in the area ofcontact between tire and road, the actual pressure distribution in thisarea must be determined.

[0257] Using the finite-element model, a static pressure distribution isdetermined on a non-rolling tire with no drift. The pressuredistribution determined under a 55 range tire, quoted earlier, isillustrated in FIG. 24, with a normal acting load of 2,914 N. Similardistributions were determined with normal loads of 4,611 N and 6,302 N.

[0258] It may be seen that two pressure peaks are presented in thetransverse direction and that these peaks are shifted outwardly withrespect to the centre of the area of contact. Furthermore thedistribution in the longitudinal direction is symmetrical. Finally theratio of the maximum value to the minimum value of the pressure alongthe transverse direction increases as the normal load increases.

[0259] To obtain the drift curves, the pressure values in those pointswhere the microinserts of the brush model are present must be knownbeforehand. The contact was represented in the concentrated-parameterphysical model by means of a regular grid of 200×15 elements (200elements longitudinally and 3×5 elements transversally). Accordingly,the pressure at the nodes of this grid must be known, i.e. at 3,000points. The finite-element model provides the pressure at a much lowernumber of points arranged in an irregular grid. A procedure for theinterpolation of the finite-element model data is used to move from thegrid of points provided by the finite-element model to that required bythe concentrated-parameter model. In the case in hand, the procedureadopted is that of the “inverse distances”. This interpolation procedurerequires the following inputs:

[0260] the coordinates of those points at which the pressure is known;

[0261] the value of the pressure at these points;

[0262] the coordinates of the points for which the interpolated pressurevalue is required.

[0263] The procedure provides the pressure value for the pointsrequired.

[0264] Downstream of the identification stage, it is required that anynegative pressure values be equal to zero. Accordingly the actual shapeof the area of contact can be reproduced extremely accurately as eachmicroinsert of the brush model has an own length and a known pressuredistribution. More specifically, the length of the contact of eachmicroinsert changes upon variation of the transversal positionconsidered and each microinsert has an own pressure value that dependson its position within the contact. Therefore the curve of the contactsof each microinsert, which identifies the position of the bottom ends ofthe microinserts when the tire is drifting, varies from one microinsertto the next. When the wheel is set rolling and drifting, and a beam andthe five rows of microinserts under it are considered, it may be seenthat the microinserts undergo a progressive, linearly increasingdeformation. The range of deformations is therefore triangular, in thepassage from entrance to exit of the footprint area, assuming there isno slipping (FIG. 25). FIG. 26 illustrates the deformation undergone bythe beam due to the effect of the sideward forces transmitted by themicroinserts and the deformations undergone by the microinserts. Thedeformations of the microinserts are given by the relative displacementsbetween the upper ends (attached to the beam) and the lower ends(located on the line of contacts in the event of adhesion). Pattern ofthe pressure in each microinsert is illustrated in FIG. 27.

[0265] In order to determine the lateral and torsional stiffness perunit of length of the microinsert, respectively {tilde over (c)}_(p) and{tilde over (k)}_(tor), the total stiffnesses of the whole tread aretaken and then the values found are divided by the total length of themicroinserts obtained as the sum of those of the single microinserts:$\begin{matrix}\left\{ \begin{matrix}{{\overset{\sim}{c}}_{p} = \frac{2 \cdot a \cdot c_{py}}{l_{tol}}} \\{{\overset{\sim}{k}}_{tor} = \frac{2{\cdot a \cdot k_{tor}}}{l_{tot}}}\end{matrix} \right. & (1.43)\end{matrix}$

[0266] where 2a is the length of the area of contact, l_(tot) is the sumof the lengths of the single microinserts, c_(py) is the stiffness perunit of length of the contact and k_(tor) is the torsional stiffness perunit of length of the contact.

[0267] These values are used to obtain the drift curves.

[0268] To determine the torsional contribution of the microinserts inthe yaw test with a non-rolling tire, the flexure of the inserts presentin the footprint area of the concentrated-parameter model is identified,the sum is obtained of the contributions of each and the value obtainedsubtracted from the torque per foot provided by the finite-elementmodel. The shape of the area of contact is provided by thefinite-element model. In this way, the purely twisting torque arisingfollowing rotation about the Z axis of the tire pressed to the ground isobtained.

[0269] The sideward force produced following deformation of eachmicroinsert and contributing to the flexure, assuming perfect adhesion,is equal to the lateral stiffness of the microinsert multiplied by therelative displacement in the plane of the contact of the top end withrespect to the bottom end of the microinsert. Assuming a rotation aboutthe centre of contact and perfect adhesion, the deformation of a genericmicroinsert is equal to d (x, y) * α′, where α′ is the rotation aboutthe Z axis effectively undergone by the tread and d is the distance ofthe microinsert taken from the centre of the contact.

[0270] The lateral stiffness of the single microinsert is determinedstarting from the stiffness per unit of length of the contact:$\begin{matrix}{{\overset{\_}{c}}_{p} = {\frac{2 \cdot a \cdot c_{py}}{4 \cdot a \cdot b} \cdot {dx} \cdot {dy}}} & (1.44)\end{matrix}$

[0271] where a is the half-length of the contact, b is the half-width ofthe contact and dx and dy are respectively the longitudinal andtransversal dimensions of each microinsert.

[0272] The bending torque due to each microinsert is calculated asfollows:

M _(f)(x, y)=F(x, y).d(x, y)   (1.45)

[0273] where F is the sideward force due to the flexure of themicroinsert and x and y are the coordinates as measured from the centreof the contact.

[0274] The total bending torque created under the area of contact forthe rotation α′ is calculated as follows: $\begin{matrix}\begin{matrix}{M_{f} = {\sum\limits_{x}{\sum\limits_{y}M_{f}}}} \\{= {\sum\limits_{x}{\sum\limits_{y}{{F\left( {x,y} \right)} \cdot {d\left( {x,y} \right)}}}}} \\{= {\sum\limits_{x}{\sum\limits_{y}{\overset{\_}{c_{p}} \cdot \alpha^{\prime} \cdot {d^{2}\left( {x,y} \right)}}}}}\end{matrix} & (1.46)\end{matrix}$

[0275] By subtracting the total bending torque from the torque per footprovided by the finite-element model, the pure twisting torque isobtained:

M _(tors) =C _(foot) −M _(f)   (1.47)

[0276] If the twisting torque is divided by the rotation α′ and by thelength of the area of contact 2a, the torsional stiffness per unit oflength of the contact is obtained: $\begin{matrix}{K_{tor} = \frac{M_{tors}}{2 \cdot a \cdot \alpha^{\prime}}} & (1.48)\end{matrix}$

[0277] With the method according to the invention making allowance forthe deformability of the structure of the concentrated-parameter model,of the local deformations of the contact through the equivalent beams,of the lateral and torsional stiffnesses of the microinserts, the driftcurves for the range 55 tire were determined for three different loads.

[0278] To obtain calculated drift curves coinciding with theexperimental ones, the shape of the area of contact is considered.

[0279] Bearing in mind the local deformation of the belt incorrespondence with the area of contact, a pear-shaped area of contactis identified and a brush model with equivalent beams of differentlengths is used, as depicted in FIG. 28. The length of the beams isobtained by performing a sensitivity analysis, assuming that the centralbeam is equal in length to the statistically measured contact arealength and that the variation of length of the external and internalbeams is of equal modulus, but opposite sign. The sensitivity analysiswas conducted on the range 55 tire.

[0280] It was found that the shape of the area of contact affects thetransversal pressure distribution: the pressure peak corresponding tothe outside of the curve is no longer equal to the other one (theexternal pressure peak increases, whereas the internal one decreases asthe drift angle increases). It is considered that the pressuredistribution in the transverse direction is always less symmetrical witha higher pressure peak in correspondence with the longest equivalentbeam and a lower pressure peak in correspondence with the shortestequivalent beam.

[0281] For an indication of the pattern of the distribution of pressurefrom the finite-element model, the sideward forces arising under thecontact are calculated for different drift angles through theconcentrated-parameter physical model as described earlier. This forcedistribution is applied to the nodes of the finite-element model(non-rolling) in contact with the ground and an extremely realisticindication of the pressure distribution in the area of contact obtained.From the pressure distributions obtained, a strong symmetry in thetransverse direction results. Illustrated in FIG. 29 is the pattern ofpressure distribution (N/mm²) for a vertical load of 2,914 N whenever asideward force and a self-aligning torque corresponding to a test withdrift of 6° are applied to the finite-element model. On application ofthese new data to the concentrated-parameter physical model, calculateddrift, sideward force and self-aligning torque curves are obtained thatare practically coincident with those obtained by experimental means.FIGS. 30 and 31 illustrate respectively the pattern of the sidewardforce (N) and the self-aligning torque (Nm) in relation to the driftangle α (°) for a vertical load of 2,914 N. The calculated values aremarked with asterisks (*), whereas the experimental values are markedwith circles (o). Similar curves were obtained for normal loads of 4,611N and 6,302 N.

[0282] To determine the dynamic behaviour of the drifting tire duringthe transient state, the tire is made roll by imposing laws of motion onthe hub that vary with time and are suitable for numerically reproducingselected experimental tests commonly carried out on tires.

[0283] Two experimental tests are conducted in the laboratory forevaluating the dynamic behaviour of a drifting tire:

[0284] a first test called pendulum test, consisting in simultaneouslyimposing a lateral displacement and a steering angle on the hub of thetire;

[0285] a second test called drift test with yaw pattern, consisting indirectly imposing a drift angle equal to the yaw angle imparted to thehub.

[0286] In both of the experimental tests, a wheel-road that simulatesthe ground is used. The tire is blown up and pressed to the wheel-roadwhich is rotating at constant angular speed. The axis of the tire ismade oscillate, and a drift angle that varies with time induced upon it.The oscillations are imposed on the wheel with the two test arrangementsdescribed above.

[0287]FIG. 32 illustrates a cinematic diagram of a test machine with thependulum arrangement. By adopting a reference system integral with thehub of the wheel, the longitudinal and transverse components are definedof the velocity of the centre of the area of footprint determined by therotation velocity V imparted to the tire by the wheel-road.

V _(T) ={dot over (σ)}.l+V. sin (σ)

V_(L)=V. cos (σ)   (1.49)

[0288] from which the drift angle α is obtained for small values of theangle of pendulum (steering) σ: $\begin{matrix}{{\alpha \cong \frac{{\overset{.}{\sigma} \cdot l} + {V \cdot \sigma}}{V}} = {\sigma + {\frac{\overset{.}{\sigma}}{V} \cdot l}}} & (1.50)\end{matrix}$

[0289] If a motion that is, for instance, sinusoidal is imposed on thependulum:

σ=σ₀. cos (Ω.t)   (1.51)

[0290] the drift angle becomes: $\begin{matrix}\begin{matrix}{\alpha = {{\sigma_{0} \cdot {\cos \left( {\Omega \cdot t} \right)}} - {\frac{\Omega \cdot \sigma_{0}}{V} \cdot l \cdot {\sin \left( {\Omega \cdot t} \right)}}}} \\{{= \sigma_{0}}{\cdot \left( {{\cos \left( {\Omega \cdot t} \right)} + \phi} \right)}}\end{matrix} & (1.52)\end{matrix}$

[0291] Therefore, the drift angle α is equal to the steering angle σimposed on the hub, with a phase difference of angle φ. If σ isconstant, then α is always=σ.

[0292]FIG. 33 illustrates a cinematic diagram of a test machine with theyaw arrangement. The longitudinal and transverse components of thevelocity of the centre of the hub are defined as follows:$\begin{matrix}\left\{ \begin{matrix}{V_{L} = {V \cdot {\cos (\sigma)}}} \\{V_{T} = {V \cdot {\sin (\sigma)}}}\end{matrix} \right. & \text{(1.53)}\end{matrix}$

[0293] The resultant drift angle for low snaking angles imposed on thehub is: $\begin{matrix}{\alpha = {\frac{V \cdot {\sin (\sigma)}}{V \cdot {\cos (\sigma)}} \cong \sigma}} & (1.54)\end{matrix}$

[0294] Again in this case the steering angle imposed on the hub is equalto the drift angle.

[0295] To determine the equations of motion of the tire under dynamicconditions, suitable for reproducing the two experimental testsdescribed above, the absolute, right-handed trio of reference axes shownin FIG. 1 is adopted. The horizontal X axis, orthogonal to the axis ofrotation of the hub, forms a null steering angle σm. In the numericalsimulations it is assumed that it is the hub that moves while thewheel-road remains motionless. The microinserts of the brush model enterthe area of contact and their bottom ends remain attached to the grounduntil slipping occurs.

[0296] The independent variables of the physical model 1 with ninedegrees of freedom are again those indicated above (1.1). The equationsof motion governing the motion of the physical model are again those inmatrix form reported above (1.7). The matrix of mass and that ofstiffness are those reported above at (1.4) and (1.5). The dampingmatrix is that given above at (1.6), wherein the terms (5,3), (5,6) and(6,5) are equal to J_(yωy) (moment of inertia and angular speed aboutthe Y axis) to take gyroscopic effects into account.

[0297] The vector of the forces comprises:

[0298] the force F_(by) and the torque M_(bz) transmitted by themicroinserts to the structure of the tire and a function of time; theresulting sideward force F_(by) is positive if orientated similarly tothe Y axis and the self-aligning torque M_(bz) is positive if orientatedsimilarly to the Z axis;

[0299] the force F_(my) and the torques M_(mx) and M_(mz) transmitted bythe test machine to the hub;

[0300] the torque M_(bx) constraining the plate not to rotate about theX axis because such a degree of freedom is not included since amonodimensional brush model is used.

F={F _(my) M _(mx) M _(mz) 0 0 0 F _(by) M _(bx) M _(bz)}^(T)   (1.55)

[0301] For the case under examination, in the concentrated-parameterphysical model in the transient state, the yaw test is simulated bystating:

y _(m)=ρ_(m)=0

σ_(m)=σ_(m)(t)   (1.56)

[0302] Then the three free coordinates of the hub are constrained in themodel whereas the roll of the brush model is not taken intoconsideration since the contact is considered to be monodimensional.

[0303] A partition is made between the generalized x _(l) andconstrained x_(v) degrees of freedom:

[0304] x={x _(l) ^(T) x _(v) ^(T)}^(T) ={{y _(c) ρ_(c) σ_(c) y _(b)σ_(b) }{y _(m) ρ_(m) σ_(m) σ_(b)}}^(T)   (1.57)

[0305] and the matrices of mass, stiffness and damping are accordinglyrearranged by performing a division into four submatrices so as toobtain two matrix equations from the equations of motion:$\begin{matrix}\left\{ \begin{matrix}{{{\left\lbrack M_{ll} \right\rbrack {\overset{¨}{\underset{\_}{x}}}_{l}} + {\left\lbrack M_{lv} \right\rbrack {\underset{\_}{\overset{¨}{x}}}_{v}} + {\left\lbrack R_{ll} \right\rbrack {\overset{.}{\underset{\_}{x}}}_{l}} + {\left\lbrack R_{lv} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{v}} + {\left\lbrack K_{ll} \right\rbrack x_{l}} + {\left\lbrack K_{lv} \right\rbrack {\underset{\_}{x}}_{v}}} = {\underset{\_}{F}}_{l}} \\{{{\left\lbrack M_{vl} \right\rbrack {\overset{¨}{\underset{\_}{x}}}_{l}} + {\left\lbrack M_{vv} \right\rbrack {\underset{\_}{\overset{¨}{x}}}_{v}} + {\left\lbrack R_{vl} \right\rbrack {\overset{.}{\underset{\_}{x}}}_{l}} + {\left\lbrack R_{vv} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{v}} + {\left\lbrack K_{vl} \right\rbrack x_{l}} + {\left\lbrack K_{vv} \right\rbrack {\underset{\_}{x}}_{v}}} = {\underset{\_}{F}}_{v}}\end{matrix} \right. & (1.58)\end{matrix}$

[0306] In the first equation, F _(l) represents the vector containingthe external active forces acting on the actual degrees of freedom; inthis case, the only external forces acting on the x _(l) are thesideward force F_(by) and the self-aligning torque M_(bz) created underthe contact and acting on the degrees of freedom of the plate. Theseforces depend on the arrangement of deformation of the inserts and onwhether or not there is local slipping; they are therefore, generallyspeaking, non-linear functions of the degrees of freedom. In the secondequation, F_(v) represents the vector of the generalized reactionsapplied to the constrained degrees of freedom.

[0307] Remembering that the vector x_(v) and its derivatives withrespect to time are vectors of known functions, the first equation maybe rewritten as follows:

[M _(ll) ]{umlaut over (x)} _(l) +[R _(ll) ]{dot over (x)} _(l) +[K_(ll) ]x _(l) =F _(l) −[M _(lv) ]{umlaut over (x)} _(v) −[R _(lv) ]{dotover (x)} _(v) −[K _(lv) ]x _(v) ={circumflex over (F)}   (1.59)

[0308] wherein the terms {circumflex over (F)} _(—) are all knownbecause they are the sum of the effective external forces and of theequivalent forces due to the motion imparted to the constraints.

[0309] The equation (1.59) therefore represents a system of “n”equations, one for each unknown x _(l); by solving this equation, themotion x_(l) of the model can be determined.

[0310] Once the equations (1.59) have been integrated and the valuesobtained for x_(l) and their derivatives, the constraining reactionsF_(v) may be obtained from the equations for the constrained degrees offreedom.

[0311] To solve the non-linear equations (1.59) with explicit numericalmethods, the procedure is to invert the matrix of mass [M_(ll)] and, asthe matrix is singular and cannot therefore be inverted, the equation(1.59) is rewritten as a first order system. A further partition is thenmade, followed by a change of variables.

[0312] The selected partition of x _(l) (1.57) is:

x _(l) ={y _(c) ρ_(c) σ_(c) y _(b) σ_(b)}^(T) ={{y _(c) ρ_(c) σ_(c) }{y_(b) σ_(b)}}^(T) ={x _(c) ^(T) x _(b) ^(T)}^(T)   (1.60)

[0313] On rearranging the already rearranged matrices of mass, stiffnessand damping as described above, and performing a division into foursubmatrices, the equation (1.59) becomes: $\begin{matrix}\left\{ \begin{matrix}{{{\left\lbrack M_{cc} \right\rbrack {\overset{¨}{\underset{\_}{x}}}_{c}} + {\left\lbrack R_{cc} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{c}} + {\left\lbrack R_{cb} \right\rbrack {\overset{.}{\underset{\_}{x}}}_{b}} + {\left\lbrack K_{cc} \right\rbrack {\underset{\_}{x}}_{c}} + {\left\lbrack K_{cb} \right\rbrack {\underset{\_}{x}}_{b}}} = {\hat{\underset{\_}{F}}}_{c}} \\{{{\left\lbrack R_{bc} \right\rbrack {\overset{.}{\underset{\_}{x}}}_{c}} + {\left\lbrack R_{bb} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{b}} + {\left\lbrack K_{bc} \right\rbrack {\underset{\_}{x}}_{c}} + {\left\lbrack K_{bb} \right\rbrack {\underset{\_}{x}}_{b}}} = {\underset{\_}{\hat{F}}}_{b}}\end{matrix} \right. & (1.61)\end{matrix}$

[0314] where {circumflex over (F)} _(c) is a vector of three elementscontaining the generalized forces acting directly on the degrees offreedom of the belt and {circumflex over (F)}_(b) is a vector of twoelements containing the generalized forces acting directly on thedegrees of freedom of the brush model (F_(by) and M_(bz)).

[0315] Finally, to obtain a first order system, an auxiliary identity isadded to the system (1.61): $\begin{matrix}\left\{ \begin{matrix}{{{\left\lbrack M_{cc} \right\rbrack {\overset{¨}{\underset{\_}{x}}}_{c}} + {\left\lbrack R_{cc} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{c}} + {\left\lbrack R_{cb} \right\rbrack {\overset{.}{\underset{\_}{x}}}_{b}} + {\left\lbrack K_{cc} \right\rbrack {\underset{\_}{x}}_{c}} + {\left\lbrack K_{cb} \right\rbrack {\underset{\_}{x}}_{b}}} = {\underset{\_}{\hat{F}}}_{c}} \\{{{\left\lbrack R_{bc} \right\rbrack {\overset{.}{\underset{\_}{x}}}_{c}} + {\left\lbrack R_{bb} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{b}} + {\left\lbrack K_{bc} \right\rbrack {\underset{\_}{x}}_{c}} + {\left\lbrack K_{bb} \right\rbrack {\underset{\_}{x}}_{b}}} = {\underset{\_}{\hat{F}}}_{b}} \\{{\left\lbrack M_{cc} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{c}} = {\left\lbrack M_{cc} \right\rbrack {\underset{\_}{\overset{.}{x}}}_{c}}}\end{matrix} \right. & (1.62)\end{matrix}$

[0316] and the following change of variables is made:

z={{dot over (x)} _(c) ^(T) x _(c) ^(T) x _(b) ^(T)}^(T) ={{{dot over(y)} _(c) {dot over (ρ)}_(c) {dot over (σ)}_(c) }{y _(c) ρ_(c) σ_(c) }{y_(b) σ_(b)}}^(T)   (1.63)

[0317] If the matrices [B] and [C] are defined: $\begin{matrix}{\lbrack B\rbrack = {{\begin{bmatrix}\left\lbrack M_{cc} \right\rbrack & \lbrack 0\rbrack & \left\lbrack R_{cb} \right\rbrack \\\lbrack 0\rbrack & \lbrack 0\rbrack & \left\lbrack R_{bb} \right\rbrack \\\lbrack 0\rbrack & \left\lbrack M_{cc} \right\rbrack & \lbrack 0\rbrack\end{bmatrix}\lbrack C\rbrack} = \begin{bmatrix}\left\lbrack R_{cc} \right\rbrack & \left\lbrack K_{cc} \right\rbrack & \left\lbrack K_{cb} \right\rbrack \\\left\lbrack R_{bc} \right\rbrack & \left\lbrack K_{bc} \right\rbrack & \left\lbrack K_{bb} \right\rbrack \\{- \left\lbrack M_{cc} \right\rbrack} & \lbrack 0\rbrack & \lbrack 0\rbrack\end{bmatrix}}} & (1.64)\end{matrix}$

[0318] the system (1.62) is synthetically expressed as:

[B].{dot over (z)}+[C].z={tilde over (F)}   (1.65)

[0319] where {circumflex over (F)} is a vector of eight elementscomprised as follows:

{tilde over (F)}={{circumflex over (F)} _(c) ^(T) {circumflex over (F)}_(b) ^(T) 0 0 0}^(T)   (1.66)

[0320] The matrix [B] is now invertible. These dynamic equations arenumerically integrated a Runge-Kutta step-by-step method of the 3^(rd)order.

[0321] In the equations (1.65), the matrices [B] and [C] are definedwhilst the vector of the forces still needs to be identified, inparticular F_(by) and M_(bz) created under the rolling tire in transientstate.

[0322] As already stated earlier, to describe the behaviour of therolling tire in the transient state, a component of displacement V*t inthe X direction is imposed on the centre of the hub. In the drift testwith yaw arrangement, small variations of the degrees of freedom areimposed on the physical model. Under these conditions, the deformationmodalities of the microinserts inside the area of contact differ fromthose under steady state conditions. An insert enters the area ofcontact in a generic position that depends on the motion of the wheeland its deformation in the contact area changes with time in a mannerdictated by the pattern of the model's degrees of freedom. The forcesarising during the transient state depend on the trajectory of the topand bottom ends of each microinsert and are, as stated, influenced bythe motion of the wheel. The deformation of a generic i-th microinsertis defined as follows:

Y _(i) _(—) _(T)(x _(i) , t)−Y _(i) _(—) _(P)(x _(i) , t)   (1.67)

[0323] where Y_(i−P) is understood to be the absolute lateraldisplacement of the top end of the i-th microinsert connected to theplate and Y_(i−T) is understood to be the absolute lateral displacementof the bottom end interacting with the ground.

[0324] A procedure enabling an instant-by-instant evaluation of thedisplacements of the top and bottom ends of all the microinserts in thecontact is now illustrated with reference to FIGS. 34 and 35. The plate21 of FIG. 1, connecting the belt to the microinserts in a genericconfiguration, is illustrated in schematic form, seen from above, inFIG. 34. Having earlier indicated the absolute lateral displacement ofthe plate with y_(b) and its absolute yaw with σ_(b), the absolutelateral displacement of the top end of the generic microinsert Y_(i−P)is:

Y _(i) _(—) _(P)(x _(i) , t)=Y _(b)(t)+(α−ξ_(l)). sin (σ_(b))   (1.68)

[0325] where ξ_(i) is the abscissa indicating the generic microinsert inthe reference integral with the wheel (FIG. 34).

[0326] The position must now be identified of the bottom ends of themicroinserts Y_(i−T) in order to determine their deformation andaccordingly obtain the contact forces.

[0327] A generic microinsert is taken at the generic instant of time atwhich it enters the contact area: the top and bottom ends, seen fromabove, are at the same point because the microinsert has not yet beendeformed. In the successive instants, the top end of the genericmicroinsert has a longitudinal displacement component V * t opposite tothe feed direction of the tire while, at the same time, due to theeffect of the absolute lateral displacement of the plate y_(b) and ofits yaw σ_(b), it also has a transverse direction component.Simultaneously, the bottom end of said microinsert, assuming perfectadhesion, remains motionless in the absolute reference system whereas,in the reference system integral with the wheel, it possesses adisplacement component V * t in the longitudinal direction. Having set adistance between two adjacent microinserts of V * dt in correspondencewith each step of integration of the dynamic equations described above,a single microinsert enters and exits from the contact (FIG. 35). Threesuccessive instants of time are taken. At the instant t, the microinsert1 has just entered under the contact and is in the deformedconfiguration. An instant later, the top end of the microinsert 1 hasgone into 1′ having a vertical coordinate ξ=V * dt, whereas the bottomend has remained in 1. Simultaneously, the non-deformed microinsert 0comes under the contact. The position occupied by the bottom end of themicroinsert 1″ (which is the third microinsert in the local reference)at the time t+2 * dt is the same as occupied by the bottom end of themicroinsert 1′ an instant earlier. In the same way, the positionoccupied by the bottom end of the microinsert 2″, the fourth in thelocal reference, at the time t+2 * dt is the same as that occupied bythe microinsert 2′ an instant earlier. The position occupied by thebottom end of the generic microinsert at the time t is:

Y _(i) _(—) _(T)(x _(i) , t)=Y _(i) _(—T) (x _(i−1) , t−dt)   (1.69)

[0328] The non-linearity of the brush model results from the possibilityof whether the microinserts slip or not. Two factors have a fundamentalrole in this: the coefficient of friction at the interface between thewheel and ground and the pressure distribution. A side view is shown ofthe reference model, at the bottom of FIG. 36. The pressure distributionis assumed to be parabolic in the longitudinal direction, whereas nopattern need be defined for the transverse direction since the brushmodel considered is monodimensional.

[0329] The normal forces per unit of length that are discharged to theground are: $\begin{matrix}{q_{z\quad \_ \quad i} = {{\frac{3 \cdot F_{z}}{4 \cdot a} \cdot \left\{ {1 - \left( \frac{x_{i}}{a} \right)^{2}} \right\} \cdot \Delta}\quad x}} & (1.70)\end{matrix}$

[0330] where F_(z) is the vertical load applied to the tire. The maximumdeformation possible for the generic microinsert is: $\begin{matrix}{y_{i\quad \_ \quad \max} = {\frac{3 \cdot F_{z} \cdot \mu}{{4 \cdot c_{p} \cdot \Delta}\quad x} \cdot \left( \frac{a^{2} - x_{i}^{2}}{a^{3}} \right)}} & (1.71)\end{matrix}$

[0331] where μ is the coefficient of friction between tire and groundand c_(py) is the stiffness per unit of length of the brush model. Whenthis maximum deformation is brought onto the line x representing the topends of the microinserts, the area within which the bottom ends mustfall is determined (seen from above, top part of FIG. 36).

[0332] Knowing the position of the top end and the bottom end of thegeneric microinsert, its deformation can be determined.

[0333] When the deformation of the microinserts and the stiffness of thetread are known, it is possible to determine the forces created underthe contact area in the Y direction.

[0334] The resulting sideward force F_(by) acting on the plate is:$\begin{matrix}{{F_{by}(t)} = {\sum\limits_{i = 1}^{n}\left( {{{Y_{i\quad \_ \quad T}\left( {x_{i},t} \right)} - {Y_{i\quad \_ \quad P}\left( {x_{i},t} \right)}}{{{\cdot c_{p}} \cdot \Delta}\quad x}} \right.}} & (1.72)\end{matrix}$

[0335] If the single sideward forces created under each singlemicroinsert are integrated and multiplied by the respective arm, theself-aligning torque M_(bz) is found as follows: $\begin{matrix}{{M_{bz}(t)} = {\sum\limits_{i = 1}^{n}\left( {{{Y_{i\quad \_ \quad T}\left( {x_{i},t} \right)} - {Y_{i\quad \_ \quad P}\left( {x_{i},t} \right)}}{{{\cdot c_{p}} \cdot \Delta}\quad {x \cdot x_{i}}}} \right.}} & (1.73)\end{matrix}$

[0336] Some results obtained for a rolling tire from the proceduredescribed are illustrated below.

[0337] To simulate the behaviour of the tire in the transient state, atest was conducted envisaging a step input of the steering angleimparted to the hub. This test is particularly important as it serves toevaluate the time taken by the tire to go into a steady state andtherefore, in practice, it gives its speed of response.

[0338] In this test, a selected steering angle is imposedinstantaneously on the hub at a time t=0 and the dynamic behaviour ofthe tire is obtained from the time history of the six free degrees offreedom of the model, of the sideward force and of the self-aligningtorque.

[0339] The test was conducted on a range 55 tire with an appliedvertical load of 2,914 N and a steering angle of four degrees imposed onthe hub. The feed speed of the tire was 30 Km/h.

[0340]FIG. 37 shows the time pattern of the lateral displacement of thebelt y_(c) with respect to the hub and of the absolute lateraldisplacement of the plate y_(b).

[0341] In FIGS. 38, 39 and 40, the time pattern is shown respectively ofthe snaking σ_(c) of the belt, of the snaking σ_(b) of the plate and ofthe roll ρ_(c) of the belt.

[0342]FIGS. 41 and 42 illustrate the time pattern respectively of thesideward force F_(by) and of the self-aligning torque M_(bz).

[0343] The pattern of the sideward force F_(by) is used to evaluate thelength of relaxation which consists of the space travelled by the tirebefore the sideward force reaches 63.2% of its steady state value.

[0344] The behaviour of the tire in drifting in the transient state isreproduced using, as the transversal drift force applied to the vehicleF_(by), that given by the following equation: $\begin{matrix}{{{\frac{\sigma}{V} \cdot {\overset{.}{F}}_{by}} + F_{by}} = {{\overset{\_}{F}}_{by}(\alpha)}} & (1.74)\end{matrix}$

[0345] where α is the instantaneous drift angle, F_(by) is the steadystate force and δ is the length of relaxation.

[0346] In order to determine the length of relaxation, the signal of thesideward force F_(by) is removed of its high frequency harmonics. Alow-pass filter was used with cut-off frequency 30 Hz (FIG. 43). Thefiltered signal is comparable to the response of a 1^(st) order systemreceiving a step signal as its input.

[0347] The general equation of motion of such a system is as follows:$\begin{matrix}{{{A \cdot \frac{F}{t}} + {B \cdot F}} = F_{external}} & (1.75)\end{matrix}$

[0348] which, through the Laplace transform, may be written as follows:$\begin{matrix}{\frac{F}{F_{external}} = {\frac{1}{{A \cdot s} + B} = {\frac{\frac{1}{B}}{{\frac{A}{B} \cdot S} + 1} = \frac{a}{{\tau \cdot s} + 1}}}} & (1.76)\end{matrix}$

[0349] where τ is the time constant of the system and represents thetime taken by the system to reach 63.2% of its steady state value. The(1.76) is the system transfer function. If a step of amplitude x₀ isprovided as input, the system response in the frequency domain is asfollows: $\begin{matrix}{{F(s)} = {\frac{x_{0}}{s} \cdot \frac{a}{{\tau \cdot s} + 1}}} & (1.77)\end{matrix}$

[0350] and, on passing to the time domain, the following is obtained:$\begin{matrix}{{F(t)} = {a \cdot x_{0} \cdot \left( {1 - ^{- \frac{t}{\tau}}} \right)}} & (1.78)\end{matrix}$

[0351] The term a * x₀ is the steady state value of F.

[0352] When the equations (1.74) and (1.75) are compared, the length ofrelaxation is obtained from the curve of the filtered sideward force.

[0353] From the filtered signal of FIG. 43, the steady state value ofthe sideward force is obtained so as to calculate 63.2% and, from here,trace back to the time constant τ. For an immediate evaluation of thesteady state value, the same filtering operation was also carried out onthe self-aligning torque (FIG. 44).

[0354] In the case in question, the relaxation length was found to be0.23 meters.

[0355] FIGS. 45-47 illustrate the results obtained with the range 55tire for three different loads and with different yaw angles imposed onthe hub with feeding speed of 100 Km/h.

[0356] Local deformability of the tire in the transient state is alsointroduced into the brush model by means of the equivalent beamdescribed above. After determining the deformation curve, the positionsof the top ends of the microinserts are identified whereas the positionof the bottom ends is determined using the procedure described above. Onthe basis of the data obtained, the single elementary forces acting oneach section of a beam can be computed (FIG. 53).

[0357] The resulting sideward force and the self-aligning torque aredetermined by integrating the forces q_(y) distributed along the beam.The constraining reactions F_(a) and F_(b) are determined in order togive the deformation curve. $\begin{matrix}{F_{y} = {\int_{- a}^{a}{q_{y}*{x}}}} & (1.79) \\{{Mz} = {\int_{- a}^{a}{q_{y}*x*{x}}}} & (1.80) \\{F_{a} = {{- \frac{F}{2}} + \frac{M_{z}}{2*a}}} & (1.81) \\{F_{b} = {{- \frac{F_{y}}{2}} - \frac{M_{z}}{2*a}}} & (1.82)\end{matrix}$

[0358] Curvature of the beam is related to the bending torque throughthe following equation:

EJ * y″(x)=+M _(f)(x)   (1.83)

[0359] This equation is resolved by determining the bending torque ineach section of the beam.

[0360]FIGS. 48, 49 and 50 illustrate the pattern respectively of thesideward force, of the self-aligning torque, of the length of relaxationat steady state in relation to the steering angle, with a rigid anddeformable contact, for the range 55 tire to which a vertical load of2,914 N is applied.

[0361]FIGS. 51 and 52 illustrate respectively the pattern of the lengthof relaxation and of the time constant in relation to velocity for therange 55 tire to which three vertical loads of 2,914 N, 4,611 N and6,302 N were applied.

1. Method for determining the road handling of a tire of a wheel for a vehicle, said tire being comprised by selected mixes of rubber and reinforcing materials, said method comprising: a) a first description of said tire by means of a first, concentrated-parameter, physical model, said first physical model comprising a rigid ring which represents the tread band provided with inserts, a belting structure and corresponding carcass portion of said tire, a disk which represents a hub of said wheel and beading of said tire, principal springs and dampers connecting said rigid ring to said hub and representing sidewalls of said tire and air under pressure inside said tire, supplementary springs and dampers representing deformation phenomena of said belting structure through the effect of a specified vertical load and a brush model simulating physical phenomena in an area of contact between said tire and a road, said area of contact having a dynamic length 2a, b) a definition of selected degrees of freedom of said first physical model, and c) an identification of equations of motion suitable for describing the motion of said first physical model under selected dynamic conditions, characterized in that it comprises d) the definition of said concentrated parameters, said concentrated parameters consisting of the mass M_(c) and a diametral moment of inertia J_(c) of said rigid ring, the mass M_(m) and a diametral moment of inertia J_(m) of said disk, structural stiffnesses K_(c) and structural dampings R_(c) respectively of said principal springs and dampers, and residual stiffnesses K_(r) and residual dampings R_(r) respectively of said supplementary springs and dampers, wherein said structural stiffnesses K_(c) consist of lateral stiffness K_(cy) between said hub and said belt, camber torsional stiffness K_(cθx) between said hub and said belt and yawing torsional stiffness K_(cθz) between said hub and said belt, said structural dampings R_(c) consist of lateral damping R_(cy) between said hub and said belt, camber torsional damping R_(cθx) between said hub and said belt and yawing torsional damping R_(cθz) between said hub and said belt, said residual stiffnesses K_(r) consist of residual lateral stiffness K_(ry), residual camber torsional stiffness K_(rθx) and residual yawing torsional stiffness K_(rθz), and said residual dampings R_(r) consist of residual lateral damping R_(ry), residual camber torsional damping R_(rθx) and residual yawing torsional damping R_(rθz), e) a description of said tire by means of a second, finite-element model comprising first elements with a selected number of nodes, suitable for describing said mixes, and second elements suitable for describing said reinforcing materials, each first finite element being associated with a first stiffness matrix which is determined by means of a selected characterization of said mixes and each second element being associated with a second supplementary stiffness matrix which is determined by means of a selected characterization of said reinforcing materials, f) a simulation on said second, finite-element model of a selected series of virtual dynamic tests for exciting said second model according to selected procedures and obtaining transfer functions and first frequency responses of selected quantities, measured at selected points of said second model, g) a description of the behaviour of said first physical model by means of equations of motion suitable for representing the above dynamic tests for obtaining second frequency responses of said selected quantities, measured at selected points of said first physical model, h) a comparison between said first and said second frequency responses of said selected quantities to determine errors that are a function of said concentrated parameters of said first physical model, and i) the identification of values for said concentrated parameters that minimize said errors so that said concentrated parameters describe the dynamic behaviour of said tire, j) the determination of selected physical quantities suitable for indicating the drift behaviour of said tire, and k) the evaluation of the drift behaviour of said tire by means of said physical quantities.
 2. Method according to claim 1, characterized in that said selected physical quantities are the total drift stiffness K_(d) of said tire, in turn comprising the structural stiffness K_(c) and the tread stiffness K_(b), and the total camber stiffness K_(γ) of said tire.
 3. Method according to claim 1, characterized in that it also comprises c) a definition of said brush model, said brush model having a stiffness per unit of length c_(py) and comprising at least one rigid plate, at least one deformable beam having a length equal to the length 2a of said area of contact and at least one microinsert associated with said beam, said microinsert consisting of at least one set of springs distributed over the entire length of said beam, said springs reproducing the uniformly distributed, lateral and torsional stiffness of said area of contact.
 4. Method according to claims 1 and 3, characterized in that said degrees of freedom referred to at previous point b) are composed of: absolute lateral displacement y_(m) of said hub, absolute yaw rotation σ_(m) of said hub and absolute rolling rotation ρ_(m) of said hub, relative lateral displacement y_(c) of said belt with respect to said hub, relative yaw rotation σ_(c) of said belt with respect to said hub and relative rolling rotation ρ_(c) of said belt with respect to said hub, absolute lateral displacement y_(b) of said plate, absolute yaw rotation σ_(b) of said plate and absolute rolling rotation ρ_(b) of said plate, and absolute lateral displacement y_(s) of the bottom ends of said at least one microinsert.
 5. Method according to the claim 1, characterized in that said selected series of virtual dynamic tests referred to at previous point f) comprises a first and a second test with said tire blown up and not pressed to the ground, said first test consisting in imposing a translation in the transverse direction y on the hub and in measuring the lateral displacement y_(c) of at least one selected cardinal point of said belt and the force created between said hub and said belt in order to identify said mass M_(c), said lateral stiffness K_(cy), and said lateral damping R_(cy), said second test consisting in imposing a camber rotation θ_(x) on said hub and in measuring the lateral displacement of at least one selected cardinal point of said belt y_(c) and the torque transmitted between said hub and said belt in order to identify said diametral moment of inertia J_(c), said camber torsional stiffness K_(cθx), said camber torsional damping R_(cθx), said yawing torsional stiffness K_(cθz) and said yawing torsional damping R_(cθz).
 6. Method according to claims 1 and 5, characterized in that said selected series of virtual dynamic tests referred to at previous point f) also comprises a third and a fourth test with said tire blown up, pressed to the ground and bereft of said tread at least in said area of contact, said third test consisting in applying to said hub a sideward force in the transverse direction F_(y) and in measuring the lateral displacement y_(c) of said hub and of at least two selected cardinal points of said belt in order to identify said residual lateral stiffness K_(ry), said residual lateral damping R_(ry), said camber residual stiffness K_(rθx), and said camber residual damping R_(rθx), said fourth test consisting in applying to said hub a yawing torque C_(θz) and in measuring the yaw rotation of said hub and the lateral displacement y_(c) of at least one selected cardinal point of said belt in order to identify said residual yawing stiffness K_(rθz) and said residual yawing damping R_(rθz).
 7. Method according to claims 1 and 3, characterized in that it also comprises m) an application to said first physical model of a drift angle α, starting from a condition in which said at least one beam is in a non-deformed configuration and said brush model has a null snaking σ_(b), n) the determination of the sideward force and the self-aligning torque that act on said hub through the effect of said drift and which depend on the difference α−σ_(b) and on the deformation of said at least one beam, o) the determination of the deformation curve of said at least one beam, p) an application of said sideward force and said self-aligning torque to said second, finite-element model in order to obtain a pressure distribution on said area of contact and q) the determination of the sideward force and the self-aligning torque that act on said hub through the effect of said drift α0 on said first physical model, that depend on the pressure distribution calculated in the previous step p), r) a check, by means of said pressure distribution obtained in the previous step p), that said sideward force and said self-aligning torque are substantially similar to those calculated in previous step q), s) a determination of the sideward force and of the self-aligning torque for said angle of drift, and t) repetition of the procedure from step m) to step s) for different values of the drift angle α to obtain drift, force and self-alignment torque curves, suitable for indicating the drift behaviour under steady state conditions of said tire, and u) the evaluation of the steady state drift behaviour of said tire.
 8. Method according to claim 1, characterized in that it also comprises i) a simulation of the behaviour of said first physical model in the drift transient state by means of equations of motion reproducing selected experimental drift tests, and ii) the determination, with a selected input of a steering angle imposed on said hub, of the pattern with time of the selected free degrees of freedom of said first physical model, of the sideward force and of the self-aligning torque in said area of contact in order to determine the length of relaxation of said tire.
 9. Method according to claim 1, characterized in that said first elements of said second, finite-element model have linear form functions and their stiffness matrix is determined by means of selected static and dynamic tests conducted on specimens of said mixes, whereas the stiffness matrix of said second elements is determined by means of selected static tests on specimens of said reinforcing materials.
 10. Tire for a wheel of a vehicle, said tire being made from selected mixes of rubber and reinforcing materials and comprising a carcass, a belting structure, a tread band provided with inserts, shoulders, sidewalls, beads provided with bead wires and bead fillings, said tire being representable by means of a first, concentrated-parameter, physical model and a brush model with a road, characterized in that said concentrated parameters comprise structural stiffnesses K_(c) consisting of lateral stiffness K_(cy), camber torsional stiffness K_(cθx) and yawing torsional stiffness K_(cθz), structural dampings R_(c) consisting of lateral damping R_(cy), camber torsional damping R_(cθx) and yawing torsional damping R_(cθz), residual stiffnesses K_(r) consisting of residual lateral stiffness K_(ry), residual camber torsional stiffness K_(rθx) and residual yawing torsional stiffness K_(rθz), and residual dampings R_(r) consisting of residual lateral damping R_(ry), residual camber torsional damping R_(rθx) and residual yawing torsional damping R_(rθz), said tire also being representable by means of a second, finite-element model comprising first elements with a selected number of nodes, suitable for describing said mixes, and second elements suitable for describing said reinforcing materials, said concentrated parameters being identified by means of a selected series of dynamic tests on said second, finite-element model and represented by equations of motion applied to said first physical model, said tire having construction characteristics substantially equivalent to said concentrated parameters which describe the dynamic behaviour of said tire and enabling the determination of selected physical quantities suitable for indicating the drift behaviour of said tire for evaluation of said tire in relation to its road handling.
 11. Tire according to claim 10, characterized in that said selected physical quantities are the total drift stiffness K_(d) of said tire, in turn comprising the structural stiffness K_(c) and the tread stiffness K_(b), and the total camber stiffness K_(γ) of said tire.
 12. Tire according to claim 11, characterized in that the total drift stiffness K_(d) and the total camber stiffness K_(γ) are within the following value ranges: K_(d)=500-2,000 [N/g] K_(γ)=40-3,500 [N/g] where g=degree.
 13. Tire according to claim 11, characterized in that the structural stiffness K_(c) and the tread stiffness K_(b) are within the following value ranges: K_(c)=8,000-30,000 [N/g] K_(b)=150-400 [N/g] where g=degree. 